src/HOL/Quotient.thy
 author Christian Sternagel Thu Aug 30 15:44:03 2012 +0900 (2012-08-30) changeset 49093 fdc301f592c4 parent 48891 c0eafbd55de3 child 51112 da97167e03f7 permissions -rw-r--r--
```     1 (*  Title:      HOL/Quotient.thy
```
```     2     Author:     Cezary Kaliszyk and Christian Urban
```
```     3 *)
```
```     4
```
```     5 header {* Definition of Quotient Types *}
```
```     6
```
```     7 theory Quotient
```
```     8 imports Plain Hilbert_Choice Equiv_Relations 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 {*
```
```    16   Basic definition for equivalence relations
```
```    17   that are represented by predicates.
```
```    18 *}
```
```    19
```
```    20 text {* Composition of Relations *}
```
```    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 subsection {* Quotient Predicate *}
```
```    32
```
```    33 definition
```
```    34   "Quotient3 R Abs Rep \<longleftrightarrow>
```
```    35      (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and>
```
```    36      (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s)"
```
```    37
```
```    38 lemma Quotient3I:
```
```    39   assumes "\<And>a. Abs (Rep a) = a"
```
```    40     and "\<And>a. R (Rep a) (Rep a)"
```
```    41     and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
```
```    42   shows "Quotient3 R Abs Rep"
```
```    43   using assms unfolding Quotient3_def by blast
```
```    44
```
```    45 lemma Quotient3_abs_rep:
```
```    46   assumes a: "Quotient3 R Abs Rep"
```
```    47   shows "Abs (Rep a) = a"
```
```    48   using a
```
```    49   unfolding Quotient3_def
```
```    50   by simp
```
```    51
```
```    52 lemma Quotient3_rep_reflp:
```
```    53   assumes a: "Quotient3 R Abs Rep"
```
```    54   shows "R (Rep a) (Rep a)"
```
```    55   using a
```
```    56   unfolding Quotient3_def
```
```    57   by blast
```
```    58
```
```    59 lemma Quotient3_rel:
```
```    60   assumes a: "Quotient3 R Abs Rep"
```
```    61   shows "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
```
```    62   using a
```
```    63   unfolding Quotient3_def
```
```    64   by blast
```
```    65
```
```    66 lemma Quotient3_refl1:
```
```    67   assumes a: "Quotient3 R Abs Rep"
```
```    68   shows "R r s \<Longrightarrow> R r r"
```
```    69   using a unfolding Quotient3_def
```
```    70   by fast
```
```    71
```
```    72 lemma Quotient3_refl2:
```
```    73   assumes a: "Quotient3 R Abs Rep"
```
```    74   shows "R r s \<Longrightarrow> R s s"
```
```    75   using a unfolding Quotient3_def
```
```    76   by fast
```
```    77
```
```    78 lemma Quotient3_rel_rep:
```
```    79   assumes a: "Quotient3 R Abs Rep"
```
```    80   shows "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
```
```    81   using a
```
```    82   unfolding Quotient3_def
```
```    83   by metis
```
```    84
```
```    85 lemma Quotient3_rep_abs:
```
```    86   assumes a: "Quotient3 R Abs Rep"
```
```    87   shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
```
```    88   using a unfolding Quotient3_def
```
```    89   by blast
```
```    90
```
```    91 lemma Quotient3_rel_abs:
```
```    92   assumes a: "Quotient3 R Abs Rep"
```
```    93   shows "R r s \<Longrightarrow> Abs r = Abs s"
```
```    94   using a unfolding Quotient3_def
```
```    95   by blast
```
```    96
```
```    97 lemma Quotient3_symp:
```
```    98   assumes a: "Quotient3 R Abs Rep"
```
```    99   shows "symp R"
```
```   100   using a unfolding Quotient3_def using sympI by metis
```
```   101
```
```   102 lemma Quotient3_transp:
```
```   103   assumes a: "Quotient3 R Abs Rep"
```
```   104   shows "transp R"
```
```   105   using a unfolding Quotient3_def using transpI by (metis (full_types))
```
```   106
```
```   107 lemma Quotient3_part_equivp:
```
```   108   assumes a: "Quotient3 R Abs Rep"
```
```   109   shows "part_equivp R"
```
```   110 by (metis Quotient3_rep_reflp Quotient3_symp Quotient3_transp a part_equivpI)
```
```   111
```
```   112 lemma identity_quotient3:
```
```   113   shows "Quotient3 (op =) id id"
```
```   114   unfolding Quotient3_def id_def
```
```   115   by blast
```
```   116
```
```   117 lemma fun_quotient3:
```
```   118   assumes q1: "Quotient3 R1 abs1 rep1"
```
```   119   and     q2: "Quotient3 R2 abs2 rep2"
```
```   120   shows "Quotient3 (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
```
```   121 proof -
```
```   122   have "\<And>a.(rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
```
```   123     using q1 q2 by (simp add: Quotient3_def fun_eq_iff)
```
```   124   moreover
```
```   125   have "\<And>a.(R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
```
```   126     by (rule fun_relI)
```
```   127       (insert q1 q2 Quotient3_rel_abs [of R1 abs1 rep1] Quotient3_rel_rep [of R2 abs2 rep2],
```
```   128         simp (no_asm) add: Quotient3_def, simp)
```
```   129
```
```   130   moreover
```
```   131   {
```
```   132   fix r s
```
```   133   have "(R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
```
```   134         (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
```
```   135   proof -
```
```   136
```
```   137     have "(R1 ===> R2) r s \<Longrightarrow> (R1 ===> R2) r r" unfolding fun_rel_def
```
```   138       using Quotient3_part_equivp[OF q1] Quotient3_part_equivp[OF q2]
```
```   139       by (metis (full_types) part_equivp_def)
```
```   140     moreover have "(R1 ===> R2) r s \<Longrightarrow> (R1 ===> R2) s s" unfolding fun_rel_def
```
```   141       using Quotient3_part_equivp[OF q1] Quotient3_part_equivp[OF q2]
```
```   142       by (metis (full_types) part_equivp_def)
```
```   143     moreover have "(R1 ===> R2) r s \<Longrightarrow> (rep1 ---> abs2) r  = (rep1 ---> abs2) s"
```
```   144       apply(auto simp add: fun_rel_def fun_eq_iff) using q1 q2 unfolding Quotient3_def by metis
```
```   145     moreover have "((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
```
```   146         (rep1 ---> abs2) r  = (rep1 ---> abs2) s) \<Longrightarrow> (R1 ===> R2) r s"
```
```   147       apply(auto simp add: fun_rel_def fun_eq_iff) using q1 q2 unfolding Quotient3_def
```
```   148     by (metis map_fun_apply)
```
```   149
```
```   150     ultimately show ?thesis by blast
```
```   151  qed
```
```   152  }
```
```   153  ultimately show ?thesis by (intro Quotient3I) (assumption+)
```
```   154 qed
```
```   155
```
```   156 lemma abs_o_rep:
```
```   157   assumes a: "Quotient3 R Abs Rep"
```
```   158   shows "Abs o Rep = id"
```
```   159   unfolding fun_eq_iff
```
```   160   by (simp add: Quotient3_abs_rep[OF a])
```
```   161
```
```   162 lemma equals_rsp:
```
```   163   assumes q: "Quotient3 R Abs Rep"
```
```   164   and     a: "R xa xb" "R ya yb"
```
```   165   shows "R xa ya = R xb yb"
```
```   166   using a Quotient3_symp[OF q] Quotient3_transp[OF q]
```
```   167   by (blast elim: sympE transpE)
```
```   168
```
```   169 lemma lambda_prs:
```
```   170   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   171   and     q2: "Quotient3 R2 Abs2 Rep2"
```
```   172   shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
```
```   173   unfolding fun_eq_iff
```
```   174   using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
```
```   175   by simp
```
```   176
```
```   177 lemma lambda_prs1:
```
```   178   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   179   and     q2: "Quotient3 R2 Abs2 Rep2"
```
```   180   shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
```
```   181   unfolding fun_eq_iff
```
```   182   using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
```
```   183   by simp
```
```   184
```
```   185 lemma rep_abs_rsp:
```
```   186   assumes q: "Quotient3 R Abs Rep"
```
```   187   and     a: "R x1 x2"
```
```   188   shows "R x1 (Rep (Abs x2))"
```
```   189   using a Quotient3_rel[OF q] Quotient3_abs_rep[OF q] Quotient3_rep_reflp[OF q]
```
```   190   by metis
```
```   191
```
```   192 lemma rep_abs_rsp_left:
```
```   193   assumes q: "Quotient3 R Abs Rep"
```
```   194   and     a: "R x1 x2"
```
```   195   shows "R (Rep (Abs x1)) x2"
```
```   196   using a Quotient3_rel[OF q] Quotient3_abs_rep[OF q] Quotient3_rep_reflp[OF q]
```
```   197   by metis
```
```   198
```
```   199 text{*
```
```   200   In the following theorem R1 can be instantiated with anything,
```
```   201   but we know some of the types of the Rep and Abs functions;
```
```   202   so by solving Quotient assumptions we can get a unique R1 that
```
```   203   will be provable; which is why we need to use @{text apply_rsp} and
```
```   204   not the primed version *}
```
```   205
```
```   206 lemma apply_rspQ3:
```
```   207   fixes f g::"'a \<Rightarrow> 'c"
```
```   208   assumes q: "Quotient3 R1 Abs1 Rep1"
```
```   209   and     a: "(R1 ===> R2) f g" "R1 x y"
```
```   210   shows "R2 (f x) (g y)"
```
```   211   using a by (auto elim: fun_relE)
```
```   212
```
```   213 lemma apply_rspQ3'':
```
```   214   assumes "Quotient3 R Abs Rep"
```
```   215   and "(R ===> S) f f"
```
```   216   shows "S (f (Rep x)) (f (Rep x))"
```
```   217 proof -
```
```   218   from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient3_rep_reflp)
```
```   219   then show ?thesis using assms(2) by (auto intro: apply_rsp')
```
```   220 qed
```
```   221
```
```   222 subsection {* lemmas for regularisation of ball and bex *}
```
```   223
```
```   224 lemma ball_reg_eqv:
```
```   225   fixes P :: "'a \<Rightarrow> bool"
```
```   226   assumes a: "equivp R"
```
```   227   shows "Ball (Respects R) P = (All P)"
```
```   228   using a
```
```   229   unfolding equivp_def
```
```   230   by (auto simp add: in_respects)
```
```   231
```
```   232 lemma bex_reg_eqv:
```
```   233   fixes P :: "'a \<Rightarrow> bool"
```
```   234   assumes a: "equivp R"
```
```   235   shows "Bex (Respects R) P = (Ex P)"
```
```   236   using a
```
```   237   unfolding equivp_def
```
```   238   by (auto simp add: in_respects)
```
```   239
```
```   240 lemma ball_reg_right:
```
```   241   assumes a: "\<And>x. x \<in> R \<Longrightarrow> P x \<longrightarrow> Q x"
```
```   242   shows "All P \<longrightarrow> Ball R Q"
```
```   243   using a by fast
```
```   244
```
```   245 lemma bex_reg_left:
```
```   246   assumes a: "\<And>x. x \<in> R \<Longrightarrow> Q x \<longrightarrow> P x"
```
```   247   shows "Bex R Q \<longrightarrow> Ex P"
```
```   248   using a by fast
```
```   249
```
```   250 lemma ball_reg_left:
```
```   251   assumes a: "equivp R"
```
```   252   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
```
```   253   using a by (metis equivp_reflp in_respects)
```
```   254
```
```   255 lemma bex_reg_right:
```
```   256   assumes a: "equivp R"
```
```   257   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
```
```   258   using a by (metis equivp_reflp in_respects)
```
```   259
```
```   260 lemma ball_reg_eqv_range:
```
```   261   fixes P::"'a \<Rightarrow> bool"
```
```   262   and x::"'a"
```
```   263   assumes a: "equivp R2"
```
```   264   shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
```
```   265   apply(rule iffI)
```
```   266   apply(rule allI)
```
```   267   apply(drule_tac x="\<lambda>y. f x" in bspec)
```
```   268   apply(simp add: in_respects fun_rel_def)
```
```   269   apply(rule impI)
```
```   270   using a equivp_reflp_symp_transp[of "R2"]
```
```   271   apply (auto elim: equivpE reflpE)
```
```   272   done
```
```   273
```
```   274 lemma bex_reg_eqv_range:
```
```   275   assumes a: "equivp R2"
```
```   276   shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
```
```   277   apply(auto)
```
```   278   apply(rule_tac x="\<lambda>y. f x" in bexI)
```
```   279   apply(simp)
```
```   280   apply(simp add: Respects_def in_respects fun_rel_def)
```
```   281   apply(rule impI)
```
```   282   using a equivp_reflp_symp_transp[of "R2"]
```
```   283   apply (auto elim: equivpE reflpE)
```
```   284   done
```
```   285
```
```   286 (* Next four lemmas are unused *)
```
```   287 lemma all_reg:
```
```   288   assumes a: "!x :: 'a. (P x --> Q x)"
```
```   289   and     b: "All P"
```
```   290   shows "All Q"
```
```   291   using a b by fast
```
```   292
```
```   293 lemma ex_reg:
```
```   294   assumes a: "!x :: 'a. (P x --> Q x)"
```
```   295   and     b: "Ex P"
```
```   296   shows "Ex Q"
```
```   297   using a b by fast
```
```   298
```
```   299 lemma ball_reg:
```
```   300   assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
```
```   301   and     b: "Ball R P"
```
```   302   shows "Ball R Q"
```
```   303   using a b by fast
```
```   304
```
```   305 lemma bex_reg:
```
```   306   assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
```
```   307   and     b: "Bex R P"
```
```   308   shows "Bex R Q"
```
```   309   using a b by fast
```
```   310
```
```   311
```
```   312 lemma ball_all_comm:
```
```   313   assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
```
```   314   shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
```
```   315   using assms by auto
```
```   316
```
```   317 lemma bex_ex_comm:
```
```   318   assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
```
```   319   shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
```
```   320   using assms by auto
```
```   321
```
```   322 subsection {* Bounded abstraction *}
```
```   323
```
```   324 definition
```
```   325   Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
```
```   326 where
```
```   327   "x \<in> p \<Longrightarrow> Babs p m x = m x"
```
```   328
```
```   329 lemma babs_rsp:
```
```   330   assumes q: "Quotient3 R1 Abs1 Rep1"
```
```   331   and     a: "(R1 ===> R2) f g"
```
```   332   shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
```
```   333   apply (auto simp add: Babs_def in_respects fun_rel_def)
```
```   334   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
```
```   335   using a apply (simp add: Babs_def fun_rel_def)
```
```   336   apply (simp add: in_respects fun_rel_def)
```
```   337   using Quotient3_rel[OF q]
```
```   338   by metis
```
```   339
```
```   340 lemma babs_prs:
```
```   341   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   342   and     q2: "Quotient3 R2 Abs2 Rep2"
```
```   343   shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
```
```   344   apply (rule ext)
```
```   345   apply (simp add:)
```
```   346   apply (subgoal_tac "Rep1 x \<in> Respects R1")
```
```   347   apply (simp add: Babs_def Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
```
```   348   apply (simp add: in_respects Quotient3_rel_rep[OF q1])
```
```   349   done
```
```   350
```
```   351 lemma babs_simp:
```
```   352   assumes q: "Quotient3 R1 Abs Rep"
```
```   353   shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
```
```   354   apply(rule iffI)
```
```   355   apply(simp_all only: babs_rsp[OF q])
```
```   356   apply(auto simp add: Babs_def fun_rel_def)
```
```   357   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
```
```   358   apply(metis Babs_def)
```
```   359   apply (simp add: in_respects)
```
```   360   using Quotient3_rel[OF q]
```
```   361   by metis
```
```   362
```
```   363 (* If a user proves that a particular functional relation
```
```   364    is an equivalence this may be useful in regularising *)
```
```   365 lemma babs_reg_eqv:
```
```   366   shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
```
```   367   by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
```
```   368
```
```   369
```
```   370 (* 3 lemmas needed for proving repabs_inj *)
```
```   371 lemma ball_rsp:
```
```   372   assumes a: "(R ===> (op =)) f g"
```
```   373   shows "Ball (Respects R) f = Ball (Respects R) g"
```
```   374   using a by (auto simp add: Ball_def in_respects elim: fun_relE)
```
```   375
```
```   376 lemma bex_rsp:
```
```   377   assumes a: "(R ===> (op =)) f g"
```
```   378   shows "(Bex (Respects R) f = Bex (Respects R) g)"
```
```   379   using a by (auto simp add: Bex_def in_respects elim: fun_relE)
```
```   380
```
```   381 lemma bex1_rsp:
```
```   382   assumes a: "(R ===> (op =)) f g"
```
```   383   shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
```
```   384   using a by (auto elim: fun_relE simp add: Ex1_def in_respects)
```
```   385
```
```   386 (* 2 lemmas needed for cleaning of quantifiers *)
```
```   387 lemma all_prs:
```
```   388   assumes a: "Quotient3 R absf repf"
```
```   389   shows "Ball (Respects R) ((absf ---> id) f) = All f"
```
```   390   using a unfolding Quotient3_def Ball_def in_respects id_apply comp_def map_fun_def
```
```   391   by metis
```
```   392
```
```   393 lemma ex_prs:
```
```   394   assumes a: "Quotient3 R absf repf"
```
```   395   shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
```
```   396   using a unfolding Quotient3_def Bex_def in_respects id_apply comp_def map_fun_def
```
```   397   by metis
```
```   398
```
```   399 subsection {* @{text Bex1_rel} quantifier *}
```
```   400
```
```   401 definition
```
```   402   Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```   403 where
```
```   404   "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)))"
```
```   405
```
```   406 lemma bex1_rel_aux:
```
```   407   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
```
```   408   unfolding Bex1_rel_def
```
```   409   apply (erule conjE)+
```
```   410   apply (erule bexE)
```
```   411   apply rule
```
```   412   apply (rule_tac x="xa" in bexI)
```
```   413   apply metis
```
```   414   apply metis
```
```   415   apply rule+
```
```   416   apply (erule_tac x="xaa" in ballE)
```
```   417   prefer 2
```
```   418   apply (metis)
```
```   419   apply (erule_tac x="ya" in ballE)
```
```   420   prefer 2
```
```   421   apply (metis)
```
```   422   apply (metis in_respects)
```
```   423   done
```
```   424
```
```   425 lemma bex1_rel_aux2:
```
```   426   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
```
```   427   unfolding Bex1_rel_def
```
```   428   apply (erule conjE)+
```
```   429   apply (erule bexE)
```
```   430   apply rule
```
```   431   apply (rule_tac x="xa" in bexI)
```
```   432   apply metis
```
```   433   apply metis
```
```   434   apply rule+
```
```   435   apply (erule_tac x="xaa" in ballE)
```
```   436   prefer 2
```
```   437   apply (metis)
```
```   438   apply (erule_tac x="ya" in ballE)
```
```   439   prefer 2
```
```   440   apply (metis)
```
```   441   apply (metis in_respects)
```
```   442   done
```
```   443
```
```   444 lemma bex1_rel_rsp:
```
```   445   assumes a: "Quotient3 R absf repf"
```
```   446   shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
```
```   447   apply (simp add: fun_rel_def)
```
```   448   apply clarify
```
```   449   apply rule
```
```   450   apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
```
```   451   apply (erule bex1_rel_aux2)
```
```   452   apply assumption
```
```   453   done
```
```   454
```
```   455
```
```   456 lemma ex1_prs:
```
```   457   assumes a: "Quotient3 R absf repf"
```
```   458   shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
```
```   459 apply (simp add:)
```
```   460 apply (subst Bex1_rel_def)
```
```   461 apply (subst Bex_def)
```
```   462 apply (subst Ex1_def)
```
```   463 apply simp
```
```   464 apply rule
```
```   465  apply (erule conjE)+
```
```   466  apply (erule_tac exE)
```
```   467  apply (erule conjE)
```
```   468  apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
```
```   469   apply (rule_tac x="absf x" in exI)
```
```   470   apply (simp)
```
```   471   apply rule+
```
```   472   using a unfolding Quotient3_def
```
```   473   apply metis
```
```   474  apply rule+
```
```   475  apply (erule_tac x="x" in ballE)
```
```   476   apply (erule_tac x="y" in ballE)
```
```   477    apply simp
```
```   478   apply (simp add: in_respects)
```
```   479  apply (simp add: in_respects)
```
```   480 apply (erule_tac exE)
```
```   481  apply rule
```
```   482  apply (rule_tac x="repf x" in exI)
```
```   483  apply (simp only: in_respects)
```
```   484   apply rule
```
```   485  apply (metis Quotient3_rel_rep[OF a])
```
```   486 using a unfolding Quotient3_def apply (simp)
```
```   487 apply rule+
```
```   488 using a unfolding Quotient3_def in_respects
```
```   489 apply metis
```
```   490 done
```
```   491
```
```   492 lemma bex1_bexeq_reg:
```
```   493   shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
```
```   494   apply (simp add: Ex1_def Bex1_rel_def in_respects)
```
```   495   apply clarify
```
```   496   apply auto
```
```   497   apply (rule bexI)
```
```   498   apply assumption
```
```   499   apply (simp add: in_respects)
```
```   500   apply (simp add: in_respects)
```
```   501   apply auto
```
```   502   done
```
```   503
```
```   504 lemma bex1_bexeq_reg_eqv:
```
```   505   assumes a: "equivp R"
```
```   506   shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
```
```   507   using equivp_reflp[OF a]
```
```   508   apply (intro impI)
```
```   509   apply (elim ex1E)
```
```   510   apply (rule mp[OF bex1_bexeq_reg])
```
```   511   apply (rule_tac a="x" in ex1I)
```
```   512   apply (subst in_respects)
```
```   513   apply (rule conjI)
```
```   514   apply assumption
```
```   515   apply assumption
```
```   516   apply clarify
```
```   517   apply (erule_tac x="xa" in allE)
```
```   518   apply simp
```
```   519   done
```
```   520
```
```   521 subsection {* Various respects and preserve lemmas *}
```
```   522
```
```   523 lemma quot_rel_rsp:
```
```   524   assumes a: "Quotient3 R Abs Rep"
```
```   525   shows "(R ===> R ===> op =) R R"
```
```   526   apply(rule fun_relI)+
```
```   527   apply(rule equals_rsp[OF a])
```
```   528   apply(assumption)+
```
```   529   done
```
```   530
```
```   531 lemma o_prs:
```
```   532   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   533   and     q2: "Quotient3 R2 Abs2 Rep2"
```
```   534   and     q3: "Quotient3 R3 Abs3 Rep3"
```
```   535   shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
```
```   536   and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
```
```   537   using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2] Quotient3_abs_rep[OF q3]
```
```   538   by (simp_all add: fun_eq_iff)
```
```   539
```
```   540 lemma o_rsp:
```
```   541   "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
```
```   542   "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
```
```   543   by (force elim: fun_relE)+
```
```   544
```
```   545 lemma cond_prs:
```
```   546   assumes a: "Quotient3 R absf repf"
```
```   547   shows "absf (if a then repf b else repf c) = (if a then b else c)"
```
```   548   using a unfolding Quotient3_def by auto
```
```   549
```
```   550 lemma if_prs:
```
```   551   assumes q: "Quotient3 R Abs Rep"
```
```   552   shows "(id ---> Rep ---> Rep ---> Abs) If = If"
```
```   553   using Quotient3_abs_rep[OF q]
```
```   554   by (auto simp add: fun_eq_iff)
```
```   555
```
```   556 lemma if_rsp:
```
```   557   assumes q: "Quotient3 R Abs Rep"
```
```   558   shows "(op = ===> R ===> R ===> R) If If"
```
```   559   by force
```
```   560
```
```   561 lemma let_prs:
```
```   562   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   563   and     q2: "Quotient3 R2 Abs2 Rep2"
```
```   564   shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
```
```   565   using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
```
```   566   by (auto simp add: fun_eq_iff)
```
```   567
```
```   568 lemma let_rsp:
```
```   569   shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
```
```   570   by (force elim: fun_relE)
```
```   571
```
```   572 lemma id_rsp:
```
```   573   shows "(R ===> R) id id"
```
```   574   by auto
```
```   575
```
```   576 lemma id_prs:
```
```   577   assumes a: "Quotient3 R Abs Rep"
```
```   578   shows "(Rep ---> Abs) id = id"
```
```   579   by (simp add: fun_eq_iff Quotient3_abs_rep [OF a])
```
```   580
```
```   581
```
```   582 locale quot_type =
```
```   583   fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   584   and   Abs :: "'a set \<Rightarrow> 'b"
```
```   585   and   Rep :: "'b \<Rightarrow> 'a set"
```
```   586   assumes equivp: "part_equivp R"
```
```   587   and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = Collect (R x)"
```
```   588   and     rep_inverse: "\<And>x. Abs (Rep x) = x"
```
```   589   and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = Collect (R x)))) \<Longrightarrow> (Rep (Abs c)) = c"
```
```   590   and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
```
```   591 begin
```
```   592
```
```   593 definition
```
```   594   abs :: "'a \<Rightarrow> 'b"
```
```   595 where
```
```   596   "abs x = Abs (Collect (R x))"
```
```   597
```
```   598 definition
```
```   599   rep :: "'b \<Rightarrow> 'a"
```
```   600 where
```
```   601   "rep a = (SOME x. x \<in> Rep a)"
```
```   602
```
```   603 lemma some_collect:
```
```   604   assumes "R r r"
```
```   605   shows "R (SOME x. x \<in> Collect (R r)) = R r"
```
```   606   apply simp
```
```   607   by (metis assms exE_some equivp[simplified part_equivp_def])
```
```   608
```
```   609 lemma Quotient:
```
```   610   shows "Quotient3 R abs rep"
```
```   611   unfolding Quotient3_def abs_def rep_def
```
```   612   proof (intro conjI allI)
```
```   613     fix a r s
```
```   614     show x: "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)" proof -
```
```   615       obtain x where r: "R x x" and rep: "Rep a = Collect (R x)" using rep_prop[of a] by auto
```
```   616       have "R (SOME x. x \<in> Rep a) x"  using r rep some_collect by metis
```
```   617       then have "R x (SOME x. x \<in> Rep a)" using part_equivp_symp[OF equivp] by fast
```
```   618       then show "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)"
```
```   619         using part_equivp_transp[OF equivp] by (metis `R (SOME x. x \<in> Rep a) x`)
```
```   620     qed
```
```   621     have "Collect (R (SOME x. x \<in> Rep a)) = (Rep a)" by (metis some_collect rep_prop)
```
```   622     then show "Abs (Collect (R (SOME x. x \<in> Rep a))) = a" using rep_inverse by auto
```
```   623     have "R r r \<Longrightarrow> R s s \<Longrightarrow> Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s"
```
```   624     proof -
```
```   625       assume "R r r" and "R s s"
```
```   626       then have "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> Collect (R r) = Collect (R s)"
```
```   627         by (metis abs_inverse)
```
```   628       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))"
```
```   629         by rule simp_all
```
```   630       finally show "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s" by simp
```
```   631     qed
```
```   632     then show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (Collect (R r)) = Abs (Collect (R s)))"
```
```   633       using equivp[simplified part_equivp_def] by metis
```
```   634     qed
```
```   635
```
```   636 end
```
```   637
```
```   638 subsection {* Quotient composition *}
```
```   639
```
```   640 lemma OOO_quotient3:
```
```   641   fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   642   fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
```
```   643   fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
```
```   644   fixes R2' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   645   fixes R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"
```
```   646   assumes R1: "Quotient3 R1 Abs1 Rep1"
```
```   647   assumes R2: "Quotient3 R2 Abs2 Rep2"
```
```   648   assumes Abs1: "\<And>x y. R2' x y \<Longrightarrow> R1 x x \<Longrightarrow> R1 y y \<Longrightarrow> R2 (Abs1 x) (Abs1 y)"
```
```   649   assumes Rep1: "\<And>x y. R2 x y \<Longrightarrow> R2' (Rep1 x) (Rep1 y)"
```
```   650   shows "Quotient3 (R1 OO R2' OO R1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
```
```   651 apply (rule Quotient3I)
```
```   652    apply (simp add: o_def Quotient3_abs_rep [OF R2] Quotient3_abs_rep [OF R1])
```
```   653   apply simp
```
```   654   apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI)
```
```   655    apply (rule Quotient3_rep_reflp [OF R1])
```
```   656   apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI [rotated])
```
```   657    apply (rule Quotient3_rep_reflp [OF R1])
```
```   658   apply (rule Rep1)
```
```   659   apply (rule Quotient3_rep_reflp [OF R2])
```
```   660  apply safe
```
```   661     apply (rename_tac x y)
```
```   662     apply (drule Abs1)
```
```   663       apply (erule Quotient3_refl2 [OF R1])
```
```   664      apply (erule Quotient3_refl1 [OF R1])
```
```   665     apply (drule Quotient3_refl1 [OF R2], drule Rep1)
```
```   666     apply (subgoal_tac "R1 r (Rep1 (Abs1 x))")
```
```   667      apply (rule_tac b="Rep1 (Abs1 x)" in relcomppI, assumption)
```
```   668      apply (erule relcomppI)
```
```   669      apply (erule Quotient3_symp [OF R1, THEN sympD])
```
```   670     apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
```
```   671     apply (rule conjI, erule Quotient3_refl1 [OF R1])
```
```   672     apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
```
```   673     apply (subst Quotient3_abs_rep [OF R1])
```
```   674     apply (erule Quotient3_rel_abs [OF R1])
```
```   675    apply (rename_tac x y)
```
```   676    apply (drule Abs1)
```
```   677      apply (erule Quotient3_refl2 [OF R1])
```
```   678     apply (erule Quotient3_refl1 [OF R1])
```
```   679    apply (drule Quotient3_refl2 [OF R2], drule Rep1)
```
```   680    apply (subgoal_tac "R1 s (Rep1 (Abs1 y))")
```
```   681     apply (rule_tac b="Rep1 (Abs1 y)" in relcomppI, assumption)
```
```   682     apply (erule relcomppI)
```
```   683     apply (erule Quotient3_symp [OF R1, THEN sympD])
```
```   684    apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
```
```   685    apply (rule conjI, erule Quotient3_refl2 [OF R1])
```
```   686    apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
```
```   687    apply (subst Quotient3_abs_rep [OF R1])
```
```   688    apply (erule Quotient3_rel_abs [OF R1, THEN sym])
```
```   689   apply simp
```
```   690   apply (rule Quotient3_rel_abs [OF R2])
```
```   691   apply (rule Quotient3_rel_abs [OF R1, THEN ssubst], assumption)
```
```   692   apply (rule Quotient3_rel_abs [OF R1, THEN subst], assumption)
```
```   693   apply (erule Abs1)
```
```   694    apply (erule Quotient3_refl2 [OF R1])
```
```   695   apply (erule Quotient3_refl1 [OF R1])
```
```   696  apply (rename_tac a b c d)
```
```   697  apply simp
```
```   698  apply (rule_tac b="Rep1 (Abs1 r)" in relcomppI)
```
```   699   apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
```
```   700   apply (rule conjI, erule Quotient3_refl1 [OF R1])
```
```   701   apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
```
```   702  apply (rule_tac b="Rep1 (Abs1 s)" in relcomppI [rotated])
```
```   703   apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
```
```   704   apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
```
```   705   apply (erule Quotient3_refl2 [OF R1])
```
```   706  apply (rule Rep1)
```
```   707  apply (drule Abs1)
```
```   708    apply (erule Quotient3_refl2 [OF R1])
```
```   709   apply (erule Quotient3_refl1 [OF R1])
```
```   710  apply (drule Abs1)
```
```   711   apply (erule Quotient3_refl2 [OF R1])
```
```   712  apply (erule Quotient3_refl1 [OF R1])
```
```   713  apply (drule Quotient3_rel_abs [OF R1])
```
```   714  apply (drule Quotient3_rel_abs [OF R1])
```
```   715  apply (drule Quotient3_rel_abs [OF R1])
```
```   716  apply (drule Quotient3_rel_abs [OF R1])
```
```   717  apply simp
```
```   718  apply (rule Quotient3_rel[symmetric, OF R2, THEN iffD2])
```
```   719  apply simp
```
```   720 done
```
```   721
```
```   722 lemma OOO_eq_quotient3:
```
```   723   fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   724   fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
```
```   725   fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
```
```   726   assumes R1: "Quotient3 R1 Abs1 Rep1"
```
```   727   assumes R2: "Quotient3 op= Abs2 Rep2"
```
```   728   shows "Quotient3 (R1 OOO op=) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
```
```   729 using assms
```
```   730 by (rule OOO_quotient3) auto
```
```   731
```
```   732 subsection {* Quotient3 to Quotient *}
```
```   733
```
```   734 lemma Quotient3_to_Quotient:
```
```   735 assumes "Quotient3 R Abs Rep"
```
```   736 and "T \<equiv> \<lambda>x y. R x x \<and> Abs x = y"
```
```   737 shows "Quotient R Abs Rep T"
```
```   738 using assms unfolding Quotient3_def by (intro QuotientI) blast+
```
```   739
```
```   740 lemma Quotient3_to_Quotient_equivp:
```
```   741 assumes q: "Quotient3 R Abs Rep"
```
```   742 and T_def: "T \<equiv> \<lambda>x y. Abs x = y"
```
```   743 and eR: "equivp R"
```
```   744 shows "Quotient R Abs Rep T"
```
```   745 proof (intro QuotientI)
```
```   746   fix a
```
```   747   show "Abs (Rep a) = a" using q by(rule Quotient3_abs_rep)
```
```   748 next
```
```   749   fix a
```
```   750   show "R (Rep a) (Rep a)" using q by(rule Quotient3_rep_reflp)
```
```   751 next
```
```   752   fix r s
```
```   753   show "R r s = (R r r \<and> R s s \<and> Abs r = Abs s)" using q by(rule Quotient3_rel[symmetric])
```
```   754 next
```
```   755   show "T = (\<lambda>x y. R x x \<and> Abs x = y)" using T_def equivp_reflp[OF eR] by simp
```
```   756 qed
```
```   757
```
```   758 subsection {* ML setup *}
```
```   759
```
```   760 text {* Auxiliary data for the quotient package *}
```
```   761
```
```   762 ML_file "Tools/Quotient/quotient_info.ML"
```
```   763 setup Quotient_Info.setup
```
```   764
```
```   765 declare [[mapQ3 "fun" = (fun_rel, fun_quotient3)]]
```
```   766
```
```   767 lemmas [quot_thm] = fun_quotient3
```
```   768 lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp id_rsp
```
```   769 lemmas [quot_preserve] = if_prs o_prs let_prs id_prs
```
```   770 lemmas [quot_equiv] = identity_equivp
```
```   771
```
```   772
```
```   773 text {* Lemmas about simplifying id's. *}
```
```   774 lemmas [id_simps] =
```
```   775   id_def[symmetric]
```
```   776   map_fun_id
```
```   777   id_apply
```
```   778   id_o
```
```   779   o_id
```
```   780   eq_comp_r
```
```   781   vimage_id
```
```   782
```
```   783 text {* Translation functions for the lifting process. *}
```
```   784 ML_file "Tools/Quotient/quotient_term.ML"
```
```   785
```
```   786
```
```   787 text {* Definitions of the quotient types. *}
```
```   788 ML_file "Tools/Quotient/quotient_type.ML"
```
```   789
```
```   790
```
```   791 text {* Definitions for quotient constants. *}
```
```   792 ML_file "Tools/Quotient/quotient_def.ML"
```
```   793
```
```   794
```
```   795 text {*
```
```   796   An auxiliary constant for recording some information
```
```   797   about the lifted theorem in a tactic.
```
```   798 *}
```
```   799 definition
```
```   800   Quot_True :: "'a \<Rightarrow> bool"
```
```   801 where
```
```   802   "Quot_True x \<longleftrightarrow> True"
```
```   803
```
```   804 lemma
```
```   805   shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
```
```   806   and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
```
```   807   and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
```
```   808   and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
```
```   809   and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
```
```   810   by (simp_all add: Quot_True_def ext)
```
```   811
```
```   812 lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
```
```   813   by (simp add: Quot_True_def)
```
```   814
```
```   815
```
```   816 text {* Tactics for proving the lifted theorems *}
```
```   817 ML_file "Tools/Quotient/quotient_tacs.ML"
```
```   818
```
```   819 subsection {* Methods / Interface *}
```
```   820
```
```   821 method_setup lifting =
```
```   822   {* Attrib.thms >> (fn thms => fn ctxt =>
```
```   823        SIMPLE_METHOD' (Quotient_Tacs.lift_tac ctxt [] thms)) *}
```
```   824   {* lift theorems to quotient types *}
```
```   825
```
```   826 method_setup lifting_setup =
```
```   827   {* Attrib.thm >> (fn thm => fn ctxt =>
```
```   828        SIMPLE_METHOD' (Quotient_Tacs.lift_procedure_tac ctxt [] thm)) *}
```
```   829   {* set up the three goals for the quotient lifting procedure *}
```
```   830
```
```   831 method_setup descending =
```
```   832   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_tac ctxt [])) *}
```
```   833   {* decend theorems to the raw level *}
```
```   834
```
```   835 method_setup descending_setup =
```
```   836   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_procedure_tac ctxt [])) *}
```
```   837   {* set up the three goals for the decending theorems *}
```
```   838
```
```   839 method_setup partiality_descending =
```
```   840   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_tac ctxt [])) *}
```
```   841   {* decend theorems to the raw level *}
```
```   842
```
```   843 method_setup partiality_descending_setup =
```
```   844   {* Scan.succeed (fn ctxt =>
```
```   845        SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_procedure_tac ctxt [])) *}
```
```   846   {* set up the three goals for the decending theorems *}
```
```   847
```
```   848 method_setup regularize =
```
```   849   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.regularize_tac ctxt)) *}
```
```   850   {* prove the regularization goals from the quotient lifting procedure *}
```
```   851
```
```   852 method_setup injection =
```
```   853   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.all_injection_tac ctxt)) *}
```
```   854   {* prove the rep/abs injection goals from the quotient lifting procedure *}
```
```   855
```
```   856 method_setup cleaning =
```
```   857   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.clean_tac ctxt)) *}
```
```   858   {* prove the cleaning goals from the quotient lifting procedure *}
```
```   859
```
```   860 attribute_setup quot_lifted =
```
```   861   {* Scan.succeed Quotient_Tacs.lifted_attrib *}
```
```   862   {* lift theorems to quotient types *}
```
```   863
```
```   864 no_notation
```
```   865   rel_conj (infixr "OOO" 75) and
```
```   866   map_fun (infixr "--->" 55) and
```
```   867   fun_rel (infixr "===>" 55)
```
```   868
```
```   869 end
```
```   870
```