src/HOL/Quotient.thy
 author kuncar Wed Apr 04 17:51:12 2012 +0200 (2012-04-04) changeset 47361 87c0eaf04bad parent 47308 9caab698dbe4 child 47362 b1f099bdfbba permissions -rw-r--r--
support non-open typedefs; define cr_rel in terms of a rep function for typedefs
```     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 uses
```
```    14   ("Tools/Quotient/quotient_info.ML")
```
```    15   ("Tools/Quotient/quotient_type.ML")
```
```    16   ("Tools/Quotient/quotient_def.ML")
```
```    17   ("Tools/Quotient/quotient_term.ML")
```
```    18   ("Tools/Quotient/quotient_tacs.ML")
```
```    19 begin
```
```    20
```
```    21 text {*
```
```    22   An aside: contravariant functorial structure of sets.
```
```    23 *}
```
```    24
```
```    25 enriched_type vimage
```
```    26   by (simp_all add: fun_eq_iff vimage_compose)
```
```    27
```
```    28 text {*
```
```    29   Basic definition for equivalence relations
```
```    30   that are represented by predicates.
```
```    31 *}
```
```    32
```
```    33 text {* Composition of Relations *}
```
```    34
```
```    35 abbreviation
```
```    36   rel_conj :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" (infixr "OOO" 75)
```
```    37 where
```
```    38   "r1 OOO r2 \<equiv> r1 OO r2 OO r1"
```
```    39
```
```    40 lemma eq_comp_r:
```
```    41   shows "((op =) OOO R) = R"
```
```    42   by (auto simp add: fun_eq_iff)
```
```    43
```
```    44 subsection {* Respects predicate *}
```
```    45
```
```    46 definition
```
```    47   Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
```
```    48 where
```
```    49   "Respects R = {x. R x x}"
```
```    50
```
```    51 lemma in_respects:
```
```    52   shows "x \<in> Respects R \<longleftrightarrow> R x x"
```
```    53   unfolding Respects_def by simp
```
```    54
```
```    55 subsection {* set map (vimage) and set relation *}
```
```    56
```
```    57 definition "set_rel R xs ys \<equiv> \<forall>x y. R x y \<longrightarrow> x \<in> xs \<longleftrightarrow> y \<in> ys"
```
```    58
```
```    59 lemma vimage_id:
```
```    60   "vimage id = id"
```
```    61   unfolding vimage_def fun_eq_iff by auto
```
```    62
```
```    63 lemma set_rel_eq:
```
```    64   "set_rel op = = op ="
```
```    65   by (subst fun_eq_iff, subst fun_eq_iff) (simp add: set_eq_iff set_rel_def)
```
```    66
```
```    67 lemma set_rel_equivp:
```
```    68   assumes e: "equivp R"
```
```    69   shows "set_rel R xs ys \<longleftrightarrow> xs = ys \<and> (\<forall>x y. x \<in> xs \<longrightarrow> R x y \<longrightarrow> y \<in> xs)"
```
```    70   unfolding set_rel_def
```
```    71   using equivp_reflp[OF e]
```
```    72   by auto (metis, metis equivp_symp[OF e])
```
```    73
```
```    74 subsection {* Quotient Predicate *}
```
```    75
```
```    76 definition
```
```    77   "Quotient3 R Abs Rep \<longleftrightarrow>
```
```    78      (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and>
```
```    79      (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s)"
```
```    80
```
```    81 lemma Quotient3I:
```
```    82   assumes "\<And>a. Abs (Rep a) = a"
```
```    83     and "\<And>a. R (Rep a) (Rep a)"
```
```    84     and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
```
```    85   shows "Quotient3 R Abs Rep"
```
```    86   using assms unfolding Quotient3_def by blast
```
```    87
```
```    88 lemma Quotient3_abs_rep:
```
```    89   assumes a: "Quotient3 R Abs Rep"
```
```    90   shows "Abs (Rep a) = a"
```
```    91   using a
```
```    92   unfolding Quotient3_def
```
```    93   by simp
```
```    94
```
```    95 lemma Quotient3_rep_reflp:
```
```    96   assumes a: "Quotient3 R Abs Rep"
```
```    97   shows "R (Rep a) (Rep a)"
```
```    98   using a
```
```    99   unfolding Quotient3_def
```
```   100   by blast
```
```   101
```
```   102 lemma Quotient3_rel:
```
```   103   assumes a: "Quotient3 R Abs Rep"
```
```   104   shows "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
```
```   105   using a
```
```   106   unfolding Quotient3_def
```
```   107   by blast
```
```   108
```
```   109 lemma Quotient3_refl1:
```
```   110   assumes a: "Quotient3 R Abs Rep"
```
```   111   shows "R r s \<Longrightarrow> R r r"
```
```   112   using a unfolding Quotient3_def
```
```   113   by fast
```
```   114
```
```   115 lemma Quotient3_refl2:
```
```   116   assumes a: "Quotient3 R Abs Rep"
```
```   117   shows "R r s \<Longrightarrow> R s s"
```
```   118   using a unfolding Quotient3_def
```
```   119   by fast
```
```   120
```
```   121 lemma Quotient3_rel_rep:
```
```   122   assumes a: "Quotient3 R Abs Rep"
```
```   123   shows "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
```
```   124   using a
```
```   125   unfolding Quotient3_def
```
```   126   by metis
```
```   127
```
```   128 lemma Quotient3_rep_abs:
```
```   129   assumes a: "Quotient3 R Abs Rep"
```
```   130   shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
```
```   131   using a unfolding Quotient3_def
```
```   132   by blast
```
```   133
```
```   134 lemma Quotient3_rel_abs:
```
```   135   assumes a: "Quotient3 R Abs Rep"
```
```   136   shows "R r s \<Longrightarrow> Abs r = Abs s"
```
```   137   using a unfolding Quotient3_def
```
```   138   by blast
```
```   139
```
```   140 lemma Quotient3_symp:
```
```   141   assumes a: "Quotient3 R Abs Rep"
```
```   142   shows "symp R"
```
```   143   using a unfolding Quotient3_def using sympI by metis
```
```   144
```
```   145 lemma Quotient3_transp:
```
```   146   assumes a: "Quotient3 R Abs Rep"
```
```   147   shows "transp R"
```
```   148   using a unfolding Quotient3_def using transpI by (metis (full_types))
```
```   149
```
```   150 lemma Quotient3_part_equivp:
```
```   151   assumes a: "Quotient3 R Abs Rep"
```
```   152   shows "part_equivp R"
```
```   153 by (metis Quotient3_rep_reflp Quotient3_symp Quotient3_transp a part_equivpI)
```
```   154
```
```   155 lemma identity_quotient3:
```
```   156   shows "Quotient3 (op =) id id"
```
```   157   unfolding Quotient3_def id_def
```
```   158   by blast
```
```   159
```
```   160 lemma fun_quotient3:
```
```   161   assumes q1: "Quotient3 R1 abs1 rep1"
```
```   162   and     q2: "Quotient3 R2 abs2 rep2"
```
```   163   shows "Quotient3 (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
```
```   164 proof -
```
```   165   have "\<And>a.(rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
```
```   166     using q1 q2 by (simp add: Quotient3_def fun_eq_iff)
```
```   167   moreover
```
```   168   have "\<And>a.(R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
```
```   169     by (rule fun_relI)
```
```   170       (insert q1 q2 Quotient3_rel_abs [of R1 abs1 rep1] Quotient3_rel_rep [of R2 abs2 rep2],
```
```   171         simp (no_asm) add: Quotient3_def, simp)
```
```   172
```
```   173   moreover
```
```   174   {
```
```   175   fix r s
```
```   176   have "(R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
```
```   177         (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
```
```   178   proof -
```
```   179
```
```   180     have "(R1 ===> R2) r s \<Longrightarrow> (R1 ===> R2) r r" unfolding fun_rel_def
```
```   181       using Quotient3_part_equivp[OF q1] Quotient3_part_equivp[OF q2]
```
```   182       by (metis (full_types) part_equivp_def)
```
```   183     moreover have "(R1 ===> R2) r s \<Longrightarrow> (R1 ===> R2) s s" unfolding fun_rel_def
```
```   184       using Quotient3_part_equivp[OF q1] Quotient3_part_equivp[OF q2]
```
```   185       by (metis (full_types) part_equivp_def)
```
```   186     moreover have "(R1 ===> R2) r s \<Longrightarrow> (rep1 ---> abs2) r  = (rep1 ---> abs2) s"
```
```   187       apply(auto simp add: fun_rel_def fun_eq_iff) using q1 q2 unfolding Quotient3_def by metis
```
```   188     moreover have "((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
```
```   189         (rep1 ---> abs2) r  = (rep1 ---> abs2) s) \<Longrightarrow> (R1 ===> R2) r s"
```
```   190       apply(auto simp add: fun_rel_def fun_eq_iff) using q1 q2 unfolding Quotient3_def
```
```   191     by (metis map_fun_apply)
```
```   192
```
```   193     ultimately show ?thesis by blast
```
```   194  qed
```
```   195  }
```
```   196  ultimately show ?thesis by (intro Quotient3I) (assumption+)
```
```   197 qed
```
```   198
```
```   199 lemma abs_o_rep:
```
```   200   assumes a: "Quotient3 R Abs Rep"
```
```   201   shows "Abs o Rep = id"
```
```   202   unfolding fun_eq_iff
```
```   203   by (simp add: Quotient3_abs_rep[OF a])
```
```   204
```
```   205 lemma equals_rsp:
```
```   206   assumes q: "Quotient3 R Abs Rep"
```
```   207   and     a: "R xa xb" "R ya yb"
```
```   208   shows "R xa ya = R xb yb"
```
```   209   using a Quotient3_symp[OF q] Quotient3_transp[OF q]
```
```   210   by (blast elim: sympE transpE)
```
```   211
```
```   212 lemma lambda_prs:
```
```   213   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   214   and     q2: "Quotient3 R2 Abs2 Rep2"
```
```   215   shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
```
```   216   unfolding fun_eq_iff
```
```   217   using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
```
```   218   by simp
```
```   219
```
```   220 lemma lambda_prs1:
```
```   221   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   222   and     q2: "Quotient3 R2 Abs2 Rep2"
```
```   223   shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
```
```   224   unfolding fun_eq_iff
```
```   225   using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
```
```   226   by simp
```
```   227
```
```   228 lemma rep_abs_rsp:
```
```   229   assumes q: "Quotient3 R Abs Rep"
```
```   230   and     a: "R x1 x2"
```
```   231   shows "R x1 (Rep (Abs x2))"
```
```   232   using a Quotient3_rel[OF q] Quotient3_abs_rep[OF q] Quotient3_rep_reflp[OF q]
```
```   233   by metis
```
```   234
```
```   235 lemma rep_abs_rsp_left:
```
```   236   assumes q: "Quotient3 R Abs Rep"
```
```   237   and     a: "R x1 x2"
```
```   238   shows "R (Rep (Abs x1)) x2"
```
```   239   using a Quotient3_rel[OF q] Quotient3_abs_rep[OF q] Quotient3_rep_reflp[OF q]
```
```   240   by metis
```
```   241
```
```   242 text{*
```
```   243   In the following theorem R1 can be instantiated with anything,
```
```   244   but we know some of the types of the Rep and Abs functions;
```
```   245   so by solving Quotient assumptions we can get a unique R1 that
```
```   246   will be provable; which is why we need to use @{text apply_rsp} and
```
```   247   not the primed version *}
```
```   248
```
```   249 lemma apply_rspQ3:
```
```   250   fixes f g::"'a \<Rightarrow> 'c"
```
```   251   assumes q: "Quotient3 R1 Abs1 Rep1"
```
```   252   and     a: "(R1 ===> R2) f g" "R1 x y"
```
```   253   shows "R2 (f x) (g y)"
```
```   254   using a by (auto elim: fun_relE)
```
```   255
```
```   256 lemma apply_rspQ3'':
```
```   257   assumes "Quotient3 R Abs Rep"
```
```   258   and "(R ===> S) f f"
```
```   259   shows "S (f (Rep x)) (f (Rep x))"
```
```   260 proof -
```
```   261   from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient3_rep_reflp)
```
```   262   then show ?thesis using assms(2) by (auto intro: apply_rsp')
```
```   263 qed
```
```   264
```
```   265 subsection {* lemmas for regularisation of ball and bex *}
```
```   266
```
```   267 lemma ball_reg_eqv:
```
```   268   fixes P :: "'a \<Rightarrow> bool"
```
```   269   assumes a: "equivp R"
```
```   270   shows "Ball (Respects R) P = (All P)"
```
```   271   using a
```
```   272   unfolding equivp_def
```
```   273   by (auto simp add: in_respects)
```
```   274
```
```   275 lemma bex_reg_eqv:
```
```   276   fixes P :: "'a \<Rightarrow> bool"
```
```   277   assumes a: "equivp R"
```
```   278   shows "Bex (Respects R) P = (Ex P)"
```
```   279   using a
```
```   280   unfolding equivp_def
```
```   281   by (auto simp add: in_respects)
```
```   282
```
```   283 lemma ball_reg_right:
```
```   284   assumes a: "\<And>x. x \<in> R \<Longrightarrow> P x \<longrightarrow> Q x"
```
```   285   shows "All P \<longrightarrow> Ball R Q"
```
```   286   using a by fast
```
```   287
```
```   288 lemma bex_reg_left:
```
```   289   assumes a: "\<And>x. x \<in> R \<Longrightarrow> Q x \<longrightarrow> P x"
```
```   290   shows "Bex R Q \<longrightarrow> Ex P"
```
```   291   using a by fast
```
```   292
```
```   293 lemma ball_reg_left:
```
```   294   assumes a: "equivp R"
```
```   295   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
```
```   296   using a by (metis equivp_reflp in_respects)
```
```   297
```
```   298 lemma bex_reg_right:
```
```   299   assumes a: "equivp R"
```
```   300   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
```
```   301   using a by (metis equivp_reflp in_respects)
```
```   302
```
```   303 lemma ball_reg_eqv_range:
```
```   304   fixes P::"'a \<Rightarrow> bool"
```
```   305   and x::"'a"
```
```   306   assumes a: "equivp R2"
```
```   307   shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
```
```   308   apply(rule iffI)
```
```   309   apply(rule allI)
```
```   310   apply(drule_tac x="\<lambda>y. f x" in bspec)
```
```   311   apply(simp add: in_respects fun_rel_def)
```
```   312   apply(rule impI)
```
```   313   using a equivp_reflp_symp_transp[of "R2"]
```
```   314   apply (auto elim: equivpE reflpE)
```
```   315   done
```
```   316
```
```   317 lemma bex_reg_eqv_range:
```
```   318   assumes a: "equivp R2"
```
```   319   shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
```
```   320   apply(auto)
```
```   321   apply(rule_tac x="\<lambda>y. f x" in bexI)
```
```   322   apply(simp)
```
```   323   apply(simp add: Respects_def in_respects fun_rel_def)
```
```   324   apply(rule impI)
```
```   325   using a equivp_reflp_symp_transp[of "R2"]
```
```   326   apply (auto elim: equivpE reflpE)
```
```   327   done
```
```   328
```
```   329 (* Next four lemmas are unused *)
```
```   330 lemma all_reg:
```
```   331   assumes a: "!x :: 'a. (P x --> Q x)"
```
```   332   and     b: "All P"
```
```   333   shows "All Q"
```
```   334   using a b by fast
```
```   335
```
```   336 lemma ex_reg:
```
```   337   assumes a: "!x :: 'a. (P x --> Q x)"
```
```   338   and     b: "Ex P"
```
```   339   shows "Ex Q"
```
```   340   using a b by fast
```
```   341
```
```   342 lemma ball_reg:
```
```   343   assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
```
```   344   and     b: "Ball R P"
```
```   345   shows "Ball R Q"
```
```   346   using a b by fast
```
```   347
```
```   348 lemma bex_reg:
```
```   349   assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
```
```   350   and     b: "Bex R P"
```
```   351   shows "Bex R Q"
```
```   352   using a b by fast
```
```   353
```
```   354
```
```   355 lemma ball_all_comm:
```
```   356   assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
```
```   357   shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
```
```   358   using assms by auto
```
```   359
```
```   360 lemma bex_ex_comm:
```
```   361   assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
```
```   362   shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
```
```   363   using assms by auto
```
```   364
```
```   365 subsection {* Bounded abstraction *}
```
```   366
```
```   367 definition
```
```   368   Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
```
```   369 where
```
```   370   "x \<in> p \<Longrightarrow> Babs p m x = m x"
```
```   371
```
```   372 lemma babs_rsp:
```
```   373   assumes q: "Quotient3 R1 Abs1 Rep1"
```
```   374   and     a: "(R1 ===> R2) f g"
```
```   375   shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
```
```   376   apply (auto simp add: Babs_def in_respects fun_rel_def)
```
```   377   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
```
```   378   using a apply (simp add: Babs_def fun_rel_def)
```
```   379   apply (simp add: in_respects fun_rel_def)
```
```   380   using Quotient3_rel[OF q]
```
```   381   by metis
```
```   382
```
```   383 lemma babs_prs:
```
```   384   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   385   and     q2: "Quotient3 R2 Abs2 Rep2"
```
```   386   shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
```
```   387   apply (rule ext)
```
```   388   apply (simp add:)
```
```   389   apply (subgoal_tac "Rep1 x \<in> Respects R1")
```
```   390   apply (simp add: Babs_def Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
```
```   391   apply (simp add: in_respects Quotient3_rel_rep[OF q1])
```
```   392   done
```
```   393
```
```   394 lemma babs_simp:
```
```   395   assumes q: "Quotient3 R1 Abs Rep"
```
```   396   shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
```
```   397   apply(rule iffI)
```
```   398   apply(simp_all only: babs_rsp[OF q])
```
```   399   apply(auto simp add: Babs_def fun_rel_def)
```
```   400   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
```
```   401   apply(metis Babs_def)
```
```   402   apply (simp add: in_respects)
```
```   403   using Quotient3_rel[OF q]
```
```   404   by metis
```
```   405
```
```   406 (* If a user proves that a particular functional relation
```
```   407    is an equivalence this may be useful in regularising *)
```
```   408 lemma babs_reg_eqv:
```
```   409   shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
```
```   410   by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
```
```   411
```
```   412
```
```   413 (* 3 lemmas needed for proving repabs_inj *)
```
```   414 lemma ball_rsp:
```
```   415   assumes a: "(R ===> (op =)) f g"
```
```   416   shows "Ball (Respects R) f = Ball (Respects R) g"
```
```   417   using a by (auto simp add: Ball_def in_respects elim: fun_relE)
```
```   418
```
```   419 lemma bex_rsp:
```
```   420   assumes a: "(R ===> (op =)) f g"
```
```   421   shows "(Bex (Respects R) f = Bex (Respects R) g)"
```
```   422   using a by (auto simp add: Bex_def in_respects elim: fun_relE)
```
```   423
```
```   424 lemma bex1_rsp:
```
```   425   assumes a: "(R ===> (op =)) f g"
```
```   426   shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
```
```   427   using a by (auto elim: fun_relE simp add: Ex1_def in_respects)
```
```   428
```
```   429 (* 2 lemmas needed for cleaning of quantifiers *)
```
```   430 lemma all_prs:
```
```   431   assumes a: "Quotient3 R absf repf"
```
```   432   shows "Ball (Respects R) ((absf ---> id) f) = All f"
```
```   433   using a unfolding Quotient3_def Ball_def in_respects id_apply comp_def map_fun_def
```
```   434   by metis
```
```   435
```
```   436 lemma ex_prs:
```
```   437   assumes a: "Quotient3 R absf repf"
```
```   438   shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
```
```   439   using a unfolding Quotient3_def Bex_def in_respects id_apply comp_def map_fun_def
```
```   440   by metis
```
```   441
```
```   442 subsection {* @{text Bex1_rel} quantifier *}
```
```   443
```
```   444 definition
```
```   445   Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```   446 where
```
```   447   "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)))"
```
```   448
```
```   449 lemma bex1_rel_aux:
```
```   450   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
```
```   451   unfolding Bex1_rel_def
```
```   452   apply (erule conjE)+
```
```   453   apply (erule bexE)
```
```   454   apply rule
```
```   455   apply (rule_tac x="xa" in bexI)
```
```   456   apply metis
```
```   457   apply metis
```
```   458   apply rule+
```
```   459   apply (erule_tac x="xaa" in ballE)
```
```   460   prefer 2
```
```   461   apply (metis)
```
```   462   apply (erule_tac x="ya" in ballE)
```
```   463   prefer 2
```
```   464   apply (metis)
```
```   465   apply (metis in_respects)
```
```   466   done
```
```   467
```
```   468 lemma bex1_rel_aux2:
```
```   469   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
```
```   470   unfolding Bex1_rel_def
```
```   471   apply (erule conjE)+
```
```   472   apply (erule bexE)
```
```   473   apply rule
```
```   474   apply (rule_tac x="xa" in bexI)
```
```   475   apply metis
```
```   476   apply metis
```
```   477   apply rule+
```
```   478   apply (erule_tac x="xaa" in ballE)
```
```   479   prefer 2
```
```   480   apply (metis)
```
```   481   apply (erule_tac x="ya" in ballE)
```
```   482   prefer 2
```
```   483   apply (metis)
```
```   484   apply (metis in_respects)
```
```   485   done
```
```   486
```
```   487 lemma bex1_rel_rsp:
```
```   488   assumes a: "Quotient3 R absf repf"
```
```   489   shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
```
```   490   apply (simp add: fun_rel_def)
```
```   491   apply clarify
```
```   492   apply rule
```
```   493   apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
```
```   494   apply (erule bex1_rel_aux2)
```
```   495   apply assumption
```
```   496   done
```
```   497
```
```   498
```
```   499 lemma ex1_prs:
```
```   500   assumes a: "Quotient3 R absf repf"
```
```   501   shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
```
```   502 apply (simp add:)
```
```   503 apply (subst Bex1_rel_def)
```
```   504 apply (subst Bex_def)
```
```   505 apply (subst Ex1_def)
```
```   506 apply simp
```
```   507 apply rule
```
```   508  apply (erule conjE)+
```
```   509  apply (erule_tac exE)
```
```   510  apply (erule conjE)
```
```   511  apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
```
```   512   apply (rule_tac x="absf x" in exI)
```
```   513   apply (simp)
```
```   514   apply rule+
```
```   515   using a unfolding Quotient3_def
```
```   516   apply metis
```
```   517  apply rule+
```
```   518  apply (erule_tac x="x" in ballE)
```
```   519   apply (erule_tac x="y" in ballE)
```
```   520    apply simp
```
```   521   apply (simp add: in_respects)
```
```   522  apply (simp add: in_respects)
```
```   523 apply (erule_tac exE)
```
```   524  apply rule
```
```   525  apply (rule_tac x="repf x" in exI)
```
```   526  apply (simp only: in_respects)
```
```   527   apply rule
```
```   528  apply (metis Quotient3_rel_rep[OF a])
```
```   529 using a unfolding Quotient3_def apply (simp)
```
```   530 apply rule+
```
```   531 using a unfolding Quotient3_def in_respects
```
```   532 apply metis
```
```   533 done
```
```   534
```
```   535 lemma bex1_bexeq_reg:
```
```   536   shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
```
```   537   apply (simp add: Ex1_def Bex1_rel_def in_respects)
```
```   538   apply clarify
```
```   539   apply auto
```
```   540   apply (rule bexI)
```
```   541   apply assumption
```
```   542   apply (simp add: in_respects)
```
```   543   apply (simp add: in_respects)
```
```   544   apply auto
```
```   545   done
```
```   546
```
```   547 lemma bex1_bexeq_reg_eqv:
```
```   548   assumes a: "equivp R"
```
```   549   shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
```
```   550   using equivp_reflp[OF a]
```
```   551   apply (intro impI)
```
```   552   apply (elim ex1E)
```
```   553   apply (rule mp[OF bex1_bexeq_reg])
```
```   554   apply (rule_tac a="x" in ex1I)
```
```   555   apply (subst in_respects)
```
```   556   apply (rule conjI)
```
```   557   apply assumption
```
```   558   apply assumption
```
```   559   apply clarify
```
```   560   apply (erule_tac x="xa" in allE)
```
```   561   apply simp
```
```   562   done
```
```   563
```
```   564 subsection {* Various respects and preserve lemmas *}
```
```   565
```
```   566 lemma quot_rel_rsp:
```
```   567   assumes a: "Quotient3 R Abs Rep"
```
```   568   shows "(R ===> R ===> op =) R R"
```
```   569   apply(rule fun_relI)+
```
```   570   apply(rule equals_rsp[OF a])
```
```   571   apply(assumption)+
```
```   572   done
```
```   573
```
```   574 lemma o_prs:
```
```   575   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   576   and     q2: "Quotient3 R2 Abs2 Rep2"
```
```   577   and     q3: "Quotient3 R3 Abs3 Rep3"
```
```   578   shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
```
```   579   and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
```
```   580   using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2] Quotient3_abs_rep[OF q3]
```
```   581   by (simp_all add: fun_eq_iff)
```
```   582
```
```   583 lemma o_rsp:
```
```   584   "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
```
```   585   "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
```
```   586   by (force elim: fun_relE)+
```
```   587
```
```   588 lemma cond_prs:
```
```   589   assumes a: "Quotient3 R absf repf"
```
```   590   shows "absf (if a then repf b else repf c) = (if a then b else c)"
```
```   591   using a unfolding Quotient3_def by auto
```
```   592
```
```   593 lemma if_prs:
```
```   594   assumes q: "Quotient3 R Abs Rep"
```
```   595   shows "(id ---> Rep ---> Rep ---> Abs) If = If"
```
```   596   using Quotient3_abs_rep[OF q]
```
```   597   by (auto simp add: fun_eq_iff)
```
```   598
```
```   599 lemma if_rsp:
```
```   600   assumes q: "Quotient3 R Abs Rep"
```
```   601   shows "(op = ===> R ===> R ===> R) If If"
```
```   602   by force
```
```   603
```
```   604 lemma let_prs:
```
```   605   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   606   and     q2: "Quotient3 R2 Abs2 Rep2"
```
```   607   shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
```
```   608   using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
```
```   609   by (auto simp add: fun_eq_iff)
```
```   610
```
```   611 lemma let_rsp:
```
```   612   shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
```
```   613   by (force elim: fun_relE)
```
```   614
```
```   615 lemma id_rsp:
```
```   616   shows "(R ===> R) id id"
```
```   617   by auto
```
```   618
```
```   619 lemma id_prs:
```
```   620   assumes a: "Quotient3 R Abs Rep"
```
```   621   shows "(Rep ---> Abs) id = id"
```
```   622   by (simp add: fun_eq_iff Quotient3_abs_rep [OF a])
```
```   623
```
```   624
```
```   625 locale quot_type =
```
```   626   fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   627   and   Abs :: "'a set \<Rightarrow> 'b"
```
```   628   and   Rep :: "'b \<Rightarrow> 'a set"
```
```   629   assumes equivp: "part_equivp R"
```
```   630   and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = Collect (R x)"
```
```   631   and     rep_inverse: "\<And>x. Abs (Rep x) = x"
```
```   632   and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = Collect (R x)))) \<Longrightarrow> (Rep (Abs c)) = c"
```
```   633   and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
```
```   634 begin
```
```   635
```
```   636 definition
```
```   637   abs :: "'a \<Rightarrow> 'b"
```
```   638 where
```
```   639   "abs x = Abs (Collect (R x))"
```
```   640
```
```   641 definition
```
```   642   rep :: "'b \<Rightarrow> 'a"
```
```   643 where
```
```   644   "rep a = (SOME x. x \<in> Rep a)"
```
```   645
```
```   646 lemma some_collect:
```
```   647   assumes "R r r"
```
```   648   shows "R (SOME x. x \<in> Collect (R r)) = R r"
```
```   649   apply simp
```
```   650   by (metis assms exE_some equivp[simplified part_equivp_def])
```
```   651
```
```   652 lemma Quotient:
```
```   653   shows "Quotient3 R abs rep"
```
```   654   unfolding Quotient3_def abs_def rep_def
```
```   655   proof (intro conjI allI)
```
```   656     fix a r s
```
```   657     show x: "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)" proof -
```
```   658       obtain x where r: "R x x" and rep: "Rep a = Collect (R x)" using rep_prop[of a] by auto
```
```   659       have "R (SOME x. x \<in> Rep a) x"  using r rep some_collect by metis
```
```   660       then have "R x (SOME x. x \<in> Rep a)" using part_equivp_symp[OF equivp] by fast
```
```   661       then show "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)"
```
```   662         using part_equivp_transp[OF equivp] by (metis `R (SOME x. x \<in> Rep a) x`)
```
```   663     qed
```
```   664     have "Collect (R (SOME x. x \<in> Rep a)) = (Rep a)" by (metis some_collect rep_prop)
```
```   665     then show "Abs (Collect (R (SOME x. x \<in> Rep a))) = a" using rep_inverse by auto
```
```   666     have "R r r \<Longrightarrow> R s s \<Longrightarrow> Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s"
```
```   667     proof -
```
```   668       assume "R r r" and "R s s"
```
```   669       then have "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> Collect (R r) = Collect (R s)"
```
```   670         by (metis abs_inverse)
```
```   671       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))"
```
```   672         by rule simp_all
```
```   673       finally show "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s" by simp
```
```   674     qed
```
```   675     then show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (Collect (R r)) = Abs (Collect (R s)))"
```
```   676       using equivp[simplified part_equivp_def] by metis
```
```   677     qed
```
```   678
```
```   679 end
```
```   680
```
```   681 subsection {* Quotient composition *}
```
```   682
```
```   683 lemma OOO_quotient3:
```
```   684   fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   685   fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
```
```   686   fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
```
```   687   fixes R2' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   688   fixes R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"
```
```   689   assumes R1: "Quotient3 R1 Abs1 Rep1"
```
```   690   assumes R2: "Quotient3 R2 Abs2 Rep2"
```
```   691   assumes Abs1: "\<And>x y. R2' x y \<Longrightarrow> R1 x x \<Longrightarrow> R1 y y \<Longrightarrow> R2 (Abs1 x) (Abs1 y)"
```
```   692   assumes Rep1: "\<And>x y. R2 x y \<Longrightarrow> R2' (Rep1 x) (Rep1 y)"
```
```   693   shows "Quotient3 (R1 OO R2' OO R1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
```
```   694 apply (rule Quotient3I)
```
```   695    apply (simp add: o_def Quotient3_abs_rep [OF R2] Quotient3_abs_rep [OF R1])
```
```   696   apply simp
```
```   697   apply (rule_tac b="Rep1 (Rep2 a)" in pred_compI)
```
```   698    apply (rule Quotient3_rep_reflp [OF R1])
```
```   699   apply (rule_tac b="Rep1 (Rep2 a)" in pred_compI [rotated])
```
```   700    apply (rule Quotient3_rep_reflp [OF R1])
```
```   701   apply (rule Rep1)
```
```   702   apply (rule Quotient3_rep_reflp [OF R2])
```
```   703  apply safe
```
```   704     apply (rename_tac x y)
```
```   705     apply (drule Abs1)
```
```   706       apply (erule Quotient3_refl2 [OF R1])
```
```   707      apply (erule Quotient3_refl1 [OF R1])
```
```   708     apply (drule Quotient3_refl1 [OF R2], drule Rep1)
```
```   709     apply (subgoal_tac "R1 r (Rep1 (Abs1 x))")
```
```   710      apply (rule_tac b="Rep1 (Abs1 x)" in pred_compI, assumption)
```
```   711      apply (erule pred_compI)
```
```   712      apply (erule Quotient3_symp [OF R1, THEN sympD])
```
```   713     apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
```
```   714     apply (rule conjI, erule Quotient3_refl1 [OF R1])
```
```   715     apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
```
```   716     apply (subst Quotient3_abs_rep [OF R1])
```
```   717     apply (erule Quotient3_rel_abs [OF R1])
```
```   718    apply (rename_tac x y)
```
```   719    apply (drule Abs1)
```
```   720      apply (erule Quotient3_refl2 [OF R1])
```
```   721     apply (erule Quotient3_refl1 [OF R1])
```
```   722    apply (drule Quotient3_refl2 [OF R2], drule Rep1)
```
```   723    apply (subgoal_tac "R1 s (Rep1 (Abs1 y))")
```
```   724     apply (rule_tac b="Rep1 (Abs1 y)" in pred_compI, assumption)
```
```   725     apply (erule pred_compI)
```
```   726     apply (erule Quotient3_symp [OF R1, THEN sympD])
```
```   727    apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
```
```   728    apply (rule conjI, erule Quotient3_refl2 [OF R1])
```
```   729    apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
```
```   730    apply (subst Quotient3_abs_rep [OF R1])
```
```   731    apply (erule Quotient3_rel_abs [OF R1, THEN sym])
```
```   732   apply simp
```
```   733   apply (rule Quotient3_rel_abs [OF R2])
```
```   734   apply (rule Quotient3_rel_abs [OF R1, THEN ssubst], assumption)
```
```   735   apply (rule Quotient3_rel_abs [OF R1, THEN subst], assumption)
```
```   736   apply (erule Abs1)
```
```   737    apply (erule Quotient3_refl2 [OF R1])
```
```   738   apply (erule Quotient3_refl1 [OF R1])
```
```   739  apply (rename_tac a b c d)
```
```   740  apply simp
```
```   741  apply (rule_tac b="Rep1 (Abs1 r)" in pred_compI)
```
```   742   apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
```
```   743   apply (rule conjI, erule Quotient3_refl1 [OF R1])
```
```   744   apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
```
```   745  apply (rule_tac b="Rep1 (Abs1 s)" in pred_compI [rotated])
```
```   746   apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
```
```   747   apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
```
```   748   apply (erule Quotient3_refl2 [OF R1])
```
```   749  apply (rule Rep1)
```
```   750  apply (drule Abs1)
```
```   751    apply (erule Quotient3_refl2 [OF R1])
```
```   752   apply (erule Quotient3_refl1 [OF R1])
```
```   753  apply (drule Abs1)
```
```   754   apply (erule Quotient3_refl2 [OF R1])
```
```   755  apply (erule Quotient3_refl1 [OF R1])
```
```   756  apply (drule Quotient3_rel_abs [OF R1])
```
```   757  apply (drule Quotient3_rel_abs [OF R1])
```
```   758  apply (drule Quotient3_rel_abs [OF R1])
```
```   759  apply (drule Quotient3_rel_abs [OF R1])
```
```   760  apply simp
```
```   761  apply (rule Quotient3_rel[symmetric, OF R2, THEN iffD2])
```
```   762  apply simp
```
```   763 done
```
```   764
```
```   765 lemma OOO_eq_quotient3:
```
```   766   fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   767   fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
```
```   768   fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
```
```   769   assumes R1: "Quotient3 R1 Abs1 Rep1"
```
```   770   assumes R2: "Quotient3 op= Abs2 Rep2"
```
```   771   shows "Quotient3 (R1 OOO op=) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
```
```   772 using assms
```
```   773 by (rule OOO_quotient3) auto
```
```   774
```
```   775 subsection {* ML setup *}
```
```   776
```
```   777 text {* Auxiliary data for the quotient package *}
```
```   778
```
```   779 use "Tools/Quotient/quotient_info.ML"
```
```   780 setup Quotient_Info.setup
```
```   781
```
```   782 declare [[mapQ3 "fun" = (fun_rel, fun_quotient3)]]
```
```   783
```
```   784 lemmas [quot_thm] = fun_quotient3
```
```   785 lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp id_rsp
```
```   786 lemmas [quot_preserve] = if_prs o_prs let_prs id_prs
```
```   787 lemmas [quot_equiv] = identity_equivp
```
```   788
```
```   789
```
```   790 text {* Lemmas about simplifying id's. *}
```
```   791 lemmas [id_simps] =
```
```   792   id_def[symmetric]
```
```   793   map_fun_id
```
```   794   id_apply
```
```   795   id_o
```
```   796   o_id
```
```   797   eq_comp_r
```
```   798   set_rel_eq
```
```   799   vimage_id
```
```   800
```
```   801 text {* Translation functions for the lifting process. *}
```
```   802 use "Tools/Quotient/quotient_term.ML"
```
```   803
```
```   804
```
```   805 text {* Definitions of the quotient types. *}
```
```   806 use "Tools/Quotient/quotient_type.ML"
```
```   807
```
```   808
```
```   809 text {* Definitions for quotient constants. *}
```
```   810 use "Tools/Quotient/quotient_def.ML"
```
```   811
```
```   812
```
```   813 text {*
```
```   814   An auxiliary constant for recording some information
```
```   815   about the lifted theorem in a tactic.
```
```   816 *}
```
```   817 definition
```
```   818   Quot_True :: "'a \<Rightarrow> bool"
```
```   819 where
```
```   820   "Quot_True x \<longleftrightarrow> True"
```
```   821
```
```   822 lemma
```
```   823   shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
```
```   824   and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
```
```   825   and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
```
```   826   and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
```
```   827   and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
```
```   828   by (simp_all add: Quot_True_def ext)
```
```   829
```
```   830 lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
```
```   831   by (simp add: Quot_True_def)
```
```   832
```
```   833
```
```   834 text {* Tactics for proving the lifted theorems *}
```
```   835 use "Tools/Quotient/quotient_tacs.ML"
```
```   836
```
```   837 subsection {* Methods / Interface *}
```
```   838
```
```   839 method_setup lifting =
```
```   840   {* Attrib.thms >> (fn thms => fn ctxt =>
```
```   841        SIMPLE_METHOD' (Quotient_Tacs.lift_tac ctxt [] thms)) *}
```
```   842   {* lift theorems to quotient types *}
```
```   843
```
```   844 method_setup lifting_setup =
```
```   845   {* Attrib.thm >> (fn thm => fn ctxt =>
```
```   846        SIMPLE_METHOD' (Quotient_Tacs.lift_procedure_tac ctxt [] thm)) *}
```
```   847   {* set up the three goals for the quotient lifting procedure *}
```
```   848
```
```   849 method_setup descending =
```
```   850   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_tac ctxt [])) *}
```
```   851   {* decend theorems to the raw level *}
```
```   852
```
```   853 method_setup descending_setup =
```
```   854   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_procedure_tac ctxt [])) *}
```
```   855   {* set up the three goals for the decending theorems *}
```
```   856
```
```   857 method_setup partiality_descending =
```
```   858   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_tac ctxt [])) *}
```
```   859   {* decend theorems to the raw level *}
```
```   860
```
```   861 method_setup partiality_descending_setup =
```
```   862   {* Scan.succeed (fn ctxt =>
```
```   863        SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_procedure_tac ctxt [])) *}
```
```   864   {* set up the three goals for the decending theorems *}
```
```   865
```
```   866 method_setup regularize =
```
```   867   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.regularize_tac ctxt)) *}
```
```   868   {* prove the regularization goals from the quotient lifting procedure *}
```
```   869
```
```   870 method_setup injection =
```
```   871   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.all_injection_tac ctxt)) *}
```
```   872   {* prove the rep/abs injection goals from the quotient lifting procedure *}
```
```   873
```
```   874 method_setup cleaning =
```
```   875   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.clean_tac ctxt)) *}
```
```   876   {* prove the cleaning goals from the quotient lifting procedure *}
```
```   877
```
```   878 attribute_setup quot_lifted =
```
```   879   {* Scan.succeed Quotient_Tacs.lifted_attrib *}
```
```   880   {* lift theorems to quotient types *}
```
```   881
```
```   882 no_notation
```
```   883   rel_conj (infixr "OOO" 75) and
```
```   884   map_fun (infixr "--->" 55) and
```
```   885   fun_rel (infixr "===>" 55)
```
```   886
```
```   887 end
```