src/HOL/Quotient.thy
 author griff Tue Apr 03 17:45:06 2012 +0900 (2012-04-03) changeset 47434 b75ce48a93ee parent 47105 e64ffc96a49f child 47435 e1b761c216ac permissions -rw-r--r--
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
```     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
```
```     9 keywords
```
```    10   "print_quotmaps" "print_quotients" "print_quotconsts" :: diag and
```
```    11   "quotient_type" :: thy_goal and "/" and
```
```    12   "setup_lifting" :: thy_decl and
```
```    13   "quotient_definition" :: thy_goal
```
```    14 uses
```
```    15   ("Tools/Quotient/quotient_info.ML")
```
```    16   ("Tools/Quotient/quotient_type.ML")
```
```    17   ("Tools/Quotient/quotient_def.ML")
```
```    18   ("Tools/Quotient/quotient_term.ML")
```
```    19   ("Tools/Quotient/quotient_tacs.ML")
```
```    20 begin
```
```    21
```
```    22 text {*
```
```    23   An aside: contravariant functorial structure of sets.
```
```    24 *}
```
```    25
```
```    26 enriched_type vimage
```
```    27   by (simp_all add: fun_eq_iff vimage_compose)
```
```    28
```
```    29 text {*
```
```    30   Basic definition for equivalence relations
```
```    31   that are represented by predicates.
```
```    32 *}
```
```    33
```
```    34 text {* Composition of Relations *}
```
```    35
```
```    36 abbreviation
```
```    37   rel_conj :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" (infixr "OOO" 75)
```
```    38 where
```
```    39   "r1 OOO r2 \<equiv> r1 OO r2 OO r1"
```
```    40
```
```    41 lemma eq_comp_r:
```
```    42   shows "((op =) OOO R) = R"
```
```    43   by (auto simp add: fun_eq_iff)
```
```    44
```
```    45 subsection {* Respects predicate *}
```
```    46
```
```    47 definition
```
```    48   Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
```
```    49 where
```
```    50   "Respects R = {x. R x x}"
```
```    51
```
```    52 lemma in_respects:
```
```    53   shows "x \<in> Respects R \<longleftrightarrow> R x x"
```
```    54   unfolding Respects_def by simp
```
```    55
```
```    56 subsection {* Function map and function relation *}
```
```    57
```
```    58 notation map_fun (infixr "--->" 55)
```
```    59
```
```    60 lemma map_fun_id:
```
```    61   "(id ---> id) = id"
```
```    62   by (simp add: fun_eq_iff)
```
```    63
```
```    64 definition
```
```    65   fun_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool" (infixr "===>" 55)
```
```    66 where
```
```    67   "fun_rel R1 R2 = (\<lambda>f g. \<forall>x y. R1 x y \<longrightarrow> R2 (f x) (g y))"
```
```    68
```
```    69 lemma fun_relI [intro]:
```
```    70   assumes "\<And>x y. R1 x y \<Longrightarrow> R2 (f x) (g y)"
```
```    71   shows "(R1 ===> R2) f g"
```
```    72   using assms by (simp add: fun_rel_def)
```
```    73
```
```    74 lemma fun_relE:
```
```    75   assumes "(R1 ===> R2) f g" and "R1 x y"
```
```    76   obtains "R2 (f x) (g y)"
```
```    77   using assms by (simp add: fun_rel_def)
```
```    78
```
```    79 lemma fun_rel_eq:
```
```    80   shows "((op =) ===> (op =)) = (op =)"
```
```    81   by (auto simp add: fun_eq_iff elim: fun_relE)
```
```    82
```
```    83 lemma fun_rel_eq_rel:
```
```    84   shows "((op =) ===> R) = (\<lambda>f g. \<forall>x. R (f x) (g x))"
```
```    85   by (simp add: fun_rel_def)
```
```    86
```
```    87 subsection {* set map (vimage) and set relation *}
```
```    88
```
```    89 definition "set_rel R xs ys \<equiv> \<forall>x y. R x y \<longrightarrow> x \<in> xs \<longleftrightarrow> y \<in> ys"
```
```    90
```
```    91 lemma vimage_id:
```
```    92   "vimage id = id"
```
```    93   unfolding vimage_def fun_eq_iff by auto
```
```    94
```
```    95 lemma set_rel_eq:
```
```    96   "set_rel op = = op ="
```
```    97   by (subst fun_eq_iff, subst fun_eq_iff) (simp add: set_eq_iff set_rel_def)
```
```    98
```
```    99 lemma set_rel_equivp:
```
```   100   assumes e: "equivp R"
```
```   101   shows "set_rel R xs ys \<longleftrightarrow> xs = ys \<and> (\<forall>x y. x \<in> xs \<longrightarrow> R x y \<longrightarrow> y \<in> xs)"
```
```   102   unfolding set_rel_def
```
```   103   using equivp_reflp[OF e]
```
```   104   by auto (metis, metis equivp_symp[OF e])
```
```   105
```
```   106 subsection {* Quotient Predicate *}
```
```   107
```
```   108 definition
```
```   109   "Quotient R Abs Rep \<longleftrightarrow>
```
```   110      (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and>
```
```   111      (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s)"
```
```   112
```
```   113 lemma QuotientI:
```
```   114   assumes "\<And>a. Abs (Rep a) = a"
```
```   115     and "\<And>a. R (Rep a) (Rep a)"
```
```   116     and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
```
```   117   shows "Quotient R Abs Rep"
```
```   118   using assms unfolding Quotient_def by blast
```
```   119
```
```   120 lemma Quotient_abs_rep:
```
```   121   assumes a: "Quotient R Abs Rep"
```
```   122   shows "Abs (Rep a) = a"
```
```   123   using a
```
```   124   unfolding Quotient_def
```
```   125   by simp
```
```   126
```
```   127 lemma Quotient_rep_reflp:
```
```   128   assumes a: "Quotient R Abs Rep"
```
```   129   shows "R (Rep a) (Rep a)"
```
```   130   using a
```
```   131   unfolding Quotient_def
```
```   132   by blast
```
```   133
```
```   134 lemma Quotient_rel:
```
```   135   assumes a: "Quotient R Abs Rep"
```
```   136   shows "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
```
```   137   using a
```
```   138   unfolding Quotient_def
```
```   139   by blast
```
```   140
```
```   141 lemma Quotient_refl1:
```
```   142   assumes a: "Quotient R Abs Rep"
```
```   143   shows "R r s \<Longrightarrow> R r r"
```
```   144   using a unfolding Quotient_def
```
```   145   by fast
```
```   146
```
```   147 lemma Quotient_refl2:
```
```   148   assumes a: "Quotient R Abs Rep"
```
```   149   shows "R r s \<Longrightarrow> R s s"
```
```   150   using a unfolding Quotient_def
```
```   151   by fast
```
```   152
```
```   153 lemma Quotient_rel_rep:
```
```   154   assumes a: "Quotient R Abs Rep"
```
```   155   shows "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
```
```   156   using a
```
```   157   unfolding Quotient_def
```
```   158   by metis
```
```   159
```
```   160 lemma Quotient_rep_abs:
```
```   161   assumes a: "Quotient R Abs Rep"
```
```   162   shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
```
```   163   using a unfolding Quotient_def
```
```   164   by blast
```
```   165
```
```   166 lemma Quotient_rel_abs:
```
```   167   assumes a: "Quotient R Abs Rep"
```
```   168   shows "R r s \<Longrightarrow> Abs r = Abs s"
```
```   169   using a unfolding Quotient_def
```
```   170   by blast
```
```   171
```
```   172 lemma Quotient_symp:
```
```   173   assumes a: "Quotient R Abs Rep"
```
```   174   shows "symp R"
```
```   175   using a unfolding Quotient_def using sympI by metis
```
```   176
```
```   177 lemma Quotient_transp:
```
```   178   assumes a: "Quotient R Abs Rep"
```
```   179   shows "transp R"
```
```   180   using a unfolding Quotient_def using transpI by metis
```
```   181
```
```   182 lemma identity_quotient:
```
```   183   shows "Quotient (op =) id id"
```
```   184   unfolding Quotient_def id_def
```
```   185   by blast
```
```   186
```
```   187 lemma fun_quotient:
```
```   188   assumes q1: "Quotient R1 abs1 rep1"
```
```   189   and     q2: "Quotient R2 abs2 rep2"
```
```   190   shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
```
```   191 proof -
```
```   192   have "\<And>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
```
```   193     using q1 q2 by (simp add: Quotient_def fun_eq_iff)
```
```   194   moreover
```
```   195   have "\<And>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
```
```   196     by (rule fun_relI)
```
```   197       (insert q1 q2 Quotient_rel_abs [of R1 abs1 rep1] Quotient_rel_rep [of R2 abs2 rep2],
```
```   198         simp (no_asm) add: Quotient_def, simp)
```
```   199   moreover
```
```   200   have "\<And>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
```
```   201         (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
```
```   202     apply(auto simp add: fun_rel_def fun_eq_iff)
```
```   203     using q1 q2 unfolding Quotient_def
```
```   204     apply(metis)
```
```   205     using q1 q2 unfolding Quotient_def
```
```   206     apply(metis)
```
```   207     using q1 q2 unfolding Quotient_def
```
```   208     apply(metis)
```
```   209     using q1 q2 unfolding Quotient_def
```
```   210     apply(metis)
```
```   211     done
```
```   212   ultimately
```
```   213   show "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
```
```   214     unfolding Quotient_def by blast
```
```   215 qed
```
```   216
```
```   217 lemma abs_o_rep:
```
```   218   assumes a: "Quotient R Abs Rep"
```
```   219   shows "Abs o Rep = id"
```
```   220   unfolding fun_eq_iff
```
```   221   by (simp add: Quotient_abs_rep[OF a])
```
```   222
```
```   223 lemma equals_rsp:
```
```   224   assumes q: "Quotient R Abs Rep"
```
```   225   and     a: "R xa xb" "R ya yb"
```
```   226   shows "R xa ya = R xb yb"
```
```   227   using a Quotient_symp[OF q] Quotient_transp[OF q]
```
```   228   by (blast elim: sympE transpE)
```
```   229
```
```   230 lemma lambda_prs:
```
```   231   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   232   and     q2: "Quotient R2 Abs2 Rep2"
```
```   233   shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
```
```   234   unfolding fun_eq_iff
```
```   235   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
```
```   236   by simp
```
```   237
```
```   238 lemma lambda_prs1:
```
```   239   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   240   and     q2: "Quotient R2 Abs2 Rep2"
```
```   241   shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
```
```   242   unfolding fun_eq_iff
```
```   243   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
```
```   244   by simp
```
```   245
```
```   246 lemma rep_abs_rsp:
```
```   247   assumes q: "Quotient R Abs Rep"
```
```   248   and     a: "R x1 x2"
```
```   249   shows "R x1 (Rep (Abs x2))"
```
```   250   using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
```
```   251   by metis
```
```   252
```
```   253 lemma rep_abs_rsp_left:
```
```   254   assumes q: "Quotient R Abs Rep"
```
```   255   and     a: "R x1 x2"
```
```   256   shows "R (Rep (Abs x1)) x2"
```
```   257   using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
```
```   258   by metis
```
```   259
```
```   260 text{*
```
```   261   In the following theorem R1 can be instantiated with anything,
```
```   262   but we know some of the types of the Rep and Abs functions;
```
```   263   so by solving Quotient assumptions we can get a unique R1 that
```
```   264   will be provable; which is why we need to use @{text apply_rsp} and
```
```   265   not the primed version *}
```
```   266
```
```   267 lemma apply_rsp:
```
```   268   fixes f g::"'a \<Rightarrow> 'c"
```
```   269   assumes q: "Quotient R1 Abs1 Rep1"
```
```   270   and     a: "(R1 ===> R2) f g" "R1 x y"
```
```   271   shows "R2 (f x) (g y)"
```
```   272   using a by (auto elim: fun_relE)
```
```   273
```
```   274 lemma apply_rsp':
```
```   275   assumes a: "(R1 ===> R2) f g" "R1 x y"
```
```   276   shows "R2 (f x) (g y)"
```
```   277   using a by (auto elim: fun_relE)
```
```   278
```
```   279 lemma apply_rsp'':
```
```   280   assumes "Quotient R Abs Rep"
```
```   281   and "(R ===> S) f f"
```
```   282   shows "S (f (Rep x)) (f (Rep x))"
```
```   283 proof -
```
```   284   from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient_rep_reflp)
```
```   285   then show ?thesis using assms(2) by (auto intro: apply_rsp')
```
```   286 qed
```
```   287
```
```   288 subsection {* lemmas for regularisation of ball and bex *}
```
```   289
```
```   290 lemma ball_reg_eqv:
```
```   291   fixes P :: "'a \<Rightarrow> bool"
```
```   292   assumes a: "equivp R"
```
```   293   shows "Ball (Respects R) P = (All P)"
```
```   294   using a
```
```   295   unfolding equivp_def
```
```   296   by (auto simp add: in_respects)
```
```   297
```
```   298 lemma bex_reg_eqv:
```
```   299   fixes P :: "'a \<Rightarrow> bool"
```
```   300   assumes a: "equivp R"
```
```   301   shows "Bex (Respects R) P = (Ex P)"
```
```   302   using a
```
```   303   unfolding equivp_def
```
```   304   by (auto simp add: in_respects)
```
```   305
```
```   306 lemma ball_reg_right:
```
```   307   assumes a: "\<And>x. x \<in> R \<Longrightarrow> P x \<longrightarrow> Q x"
```
```   308   shows "All P \<longrightarrow> Ball R Q"
```
```   309   using a by fast
```
```   310
```
```   311 lemma bex_reg_left:
```
```   312   assumes a: "\<And>x. x \<in> R \<Longrightarrow> Q x \<longrightarrow> P x"
```
```   313   shows "Bex R Q \<longrightarrow> Ex P"
```
```   314   using a by fast
```
```   315
```
```   316 lemma ball_reg_left:
```
```   317   assumes a: "equivp R"
```
```   318   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
```
```   319   using a by (metis equivp_reflp in_respects)
```
```   320
```
```   321 lemma bex_reg_right:
```
```   322   assumes a: "equivp R"
```
```   323   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
```
```   324   using a by (metis equivp_reflp in_respects)
```
```   325
```
```   326 lemma ball_reg_eqv_range:
```
```   327   fixes P::"'a \<Rightarrow> bool"
```
```   328   and x::"'a"
```
```   329   assumes a: "equivp R2"
```
```   330   shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
```
```   331   apply(rule iffI)
```
```   332   apply(rule allI)
```
```   333   apply(drule_tac x="\<lambda>y. f x" in bspec)
```
```   334   apply(simp add: in_respects fun_rel_def)
```
```   335   apply(rule impI)
```
```   336   using a equivp_reflp_symp_transp[of "R2"]
```
```   337   apply (auto elim: equivpE reflpE)
```
```   338   done
```
```   339
```
```   340 lemma bex_reg_eqv_range:
```
```   341   assumes a: "equivp R2"
```
```   342   shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
```
```   343   apply(auto)
```
```   344   apply(rule_tac x="\<lambda>y. f x" in bexI)
```
```   345   apply(simp)
```
```   346   apply(simp add: Respects_def in_respects fun_rel_def)
```
```   347   apply(rule impI)
```
```   348   using a equivp_reflp_symp_transp[of "R2"]
```
```   349   apply (auto elim: equivpE reflpE)
```
```   350   done
```
```   351
```
```   352 (* Next four lemmas are unused *)
```
```   353 lemma all_reg:
```
```   354   assumes a: "!x :: 'a. (P x --> Q x)"
```
```   355   and     b: "All P"
```
```   356   shows "All Q"
```
```   357   using a b by fast
```
```   358
```
```   359 lemma ex_reg:
```
```   360   assumes a: "!x :: 'a. (P x --> Q x)"
```
```   361   and     b: "Ex P"
```
```   362   shows "Ex Q"
```
```   363   using a b by fast
```
```   364
```
```   365 lemma ball_reg:
```
```   366   assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
```
```   367   and     b: "Ball R P"
```
```   368   shows "Ball R Q"
```
```   369   using a b by fast
```
```   370
```
```   371 lemma bex_reg:
```
```   372   assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
```
```   373   and     b: "Bex R P"
```
```   374   shows "Bex R Q"
```
```   375   using a b by fast
```
```   376
```
```   377
```
```   378 lemma ball_all_comm:
```
```   379   assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
```
```   380   shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
```
```   381   using assms by auto
```
```   382
```
```   383 lemma bex_ex_comm:
```
```   384   assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
```
```   385   shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
```
```   386   using assms by auto
```
```   387
```
```   388 subsection {* Bounded abstraction *}
```
```   389
```
```   390 definition
```
```   391   Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
```
```   392 where
```
```   393   "x \<in> p \<Longrightarrow> Babs p m x = m x"
```
```   394
```
```   395 lemma babs_rsp:
```
```   396   assumes q: "Quotient R1 Abs1 Rep1"
```
```   397   and     a: "(R1 ===> R2) f g"
```
```   398   shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
```
```   399   apply (auto simp add: Babs_def in_respects fun_rel_def)
```
```   400   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
```
```   401   using a apply (simp add: Babs_def fun_rel_def)
```
```   402   apply (simp add: in_respects fun_rel_def)
```
```   403   using Quotient_rel[OF q]
```
```   404   by metis
```
```   405
```
```   406 lemma babs_prs:
```
```   407   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   408   and     q2: "Quotient R2 Abs2 Rep2"
```
```   409   shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
```
```   410   apply (rule ext)
```
```   411   apply (simp add:)
```
```   412   apply (subgoal_tac "Rep1 x \<in> Respects R1")
```
```   413   apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
```
```   414   apply (simp add: in_respects Quotient_rel_rep[OF q1])
```
```   415   done
```
```   416
```
```   417 lemma babs_simp:
```
```   418   assumes q: "Quotient R1 Abs Rep"
```
```   419   shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
```
```   420   apply(rule iffI)
```
```   421   apply(simp_all only: babs_rsp[OF q])
```
```   422   apply(auto simp add: Babs_def fun_rel_def)
```
```   423   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
```
```   424   apply(metis Babs_def)
```
```   425   apply (simp add: in_respects)
```
```   426   using Quotient_rel[OF q]
```
```   427   by metis
```
```   428
```
```   429 (* If a user proves that a particular functional relation
```
```   430    is an equivalence this may be useful in regularising *)
```
```   431 lemma babs_reg_eqv:
```
```   432   shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
```
```   433   by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
```
```   434
```
```   435
```
```   436 (* 3 lemmas needed for proving repabs_inj *)
```
```   437 lemma ball_rsp:
```
```   438   assumes a: "(R ===> (op =)) f g"
```
```   439   shows "Ball (Respects R) f = Ball (Respects R) g"
```
```   440   using a by (auto simp add: Ball_def in_respects elim: fun_relE)
```
```   441
```
```   442 lemma bex_rsp:
```
```   443   assumes a: "(R ===> (op =)) f g"
```
```   444   shows "(Bex (Respects R) f = Bex (Respects R) g)"
```
```   445   using a by (auto simp add: Bex_def in_respects elim: fun_relE)
```
```   446
```
```   447 lemma bex1_rsp:
```
```   448   assumes a: "(R ===> (op =)) f g"
```
```   449   shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
```
```   450   using a by (auto elim: fun_relE simp add: Ex1_def in_respects)
```
```   451
```
```   452 (* 2 lemmas needed for cleaning of quantifiers *)
```
```   453 lemma all_prs:
```
```   454   assumes a: "Quotient R absf repf"
```
```   455   shows "Ball (Respects R) ((absf ---> id) f) = All f"
```
```   456   using a unfolding Quotient_def Ball_def in_respects id_apply comp_def map_fun_def
```
```   457   by metis
```
```   458
```
```   459 lemma ex_prs:
```
```   460   assumes a: "Quotient R absf repf"
```
```   461   shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
```
```   462   using a unfolding Quotient_def Bex_def in_respects id_apply comp_def map_fun_def
```
```   463   by metis
```
```   464
```
```   465 subsection {* @{text Bex1_rel} quantifier *}
```
```   466
```
```   467 definition
```
```   468   Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```   469 where
```
```   470   "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)))"
```
```   471
```
```   472 lemma bex1_rel_aux:
```
```   473   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
```
```   474   unfolding Bex1_rel_def
```
```   475   apply (erule conjE)+
```
```   476   apply (erule bexE)
```
```   477   apply rule
```
```   478   apply (rule_tac x="xa" in bexI)
```
```   479   apply metis
```
```   480   apply metis
```
```   481   apply rule+
```
```   482   apply (erule_tac x="xaa" in ballE)
```
```   483   prefer 2
```
```   484   apply (metis)
```
```   485   apply (erule_tac x="ya" in ballE)
```
```   486   prefer 2
```
```   487   apply (metis)
```
```   488   apply (metis in_respects)
```
```   489   done
```
```   490
```
```   491 lemma bex1_rel_aux2:
```
```   492   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
```
```   493   unfolding Bex1_rel_def
```
```   494   apply (erule conjE)+
```
```   495   apply (erule bexE)
```
```   496   apply rule
```
```   497   apply (rule_tac x="xa" in bexI)
```
```   498   apply metis
```
```   499   apply metis
```
```   500   apply rule+
```
```   501   apply (erule_tac x="xaa" in ballE)
```
```   502   prefer 2
```
```   503   apply (metis)
```
```   504   apply (erule_tac x="ya" in ballE)
```
```   505   prefer 2
```
```   506   apply (metis)
```
```   507   apply (metis in_respects)
```
```   508   done
```
```   509
```
```   510 lemma bex1_rel_rsp:
```
```   511   assumes a: "Quotient R absf repf"
```
```   512   shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
```
```   513   apply (simp add: fun_rel_def)
```
```   514   apply clarify
```
```   515   apply rule
```
```   516   apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
```
```   517   apply (erule bex1_rel_aux2)
```
```   518   apply assumption
```
```   519   done
```
```   520
```
```   521
```
```   522 lemma ex1_prs:
```
```   523   assumes a: "Quotient R absf repf"
```
```   524   shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
```
```   525 apply (simp add:)
```
```   526 apply (subst Bex1_rel_def)
```
```   527 apply (subst Bex_def)
```
```   528 apply (subst Ex1_def)
```
```   529 apply simp
```
```   530 apply rule
```
```   531  apply (erule conjE)+
```
```   532  apply (erule_tac exE)
```
```   533  apply (erule conjE)
```
```   534  apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
```
```   535   apply (rule_tac x="absf x" in exI)
```
```   536   apply (simp)
```
```   537   apply rule+
```
```   538   using a unfolding Quotient_def
```
```   539   apply metis
```
```   540  apply rule+
```
```   541  apply (erule_tac x="x" in ballE)
```
```   542   apply (erule_tac x="y" in ballE)
```
```   543    apply simp
```
```   544   apply (simp add: in_respects)
```
```   545  apply (simp add: in_respects)
```
```   546 apply (erule_tac exE)
```
```   547  apply rule
```
```   548  apply (rule_tac x="repf x" in exI)
```
```   549  apply (simp only: in_respects)
```
```   550   apply rule
```
```   551  apply (metis Quotient_rel_rep[OF a])
```
```   552 using a unfolding Quotient_def apply (simp)
```
```   553 apply rule+
```
```   554 using a unfolding Quotient_def in_respects
```
```   555 apply metis
```
```   556 done
```
```   557
```
```   558 lemma bex1_bexeq_reg:
```
```   559   shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
```
```   560   apply (simp add: Ex1_def Bex1_rel_def in_respects)
```
```   561   apply clarify
```
```   562   apply auto
```
```   563   apply (rule bexI)
```
```   564   apply assumption
```
```   565   apply (simp add: in_respects)
```
```   566   apply (simp add: in_respects)
```
```   567   apply auto
```
```   568   done
```
```   569
```
```   570 lemma bex1_bexeq_reg_eqv:
```
```   571   assumes a: "equivp R"
```
```   572   shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
```
```   573   using equivp_reflp[OF a]
```
```   574   apply (intro impI)
```
```   575   apply (elim ex1E)
```
```   576   apply (rule mp[OF bex1_bexeq_reg])
```
```   577   apply (rule_tac a="x" in ex1I)
```
```   578   apply (subst in_respects)
```
```   579   apply (rule conjI)
```
```   580   apply assumption
```
```   581   apply assumption
```
```   582   apply clarify
```
```   583   apply (erule_tac x="xa" in allE)
```
```   584   apply simp
```
```   585   done
```
```   586
```
```   587 subsection {* Various respects and preserve lemmas *}
```
```   588
```
```   589 lemma quot_rel_rsp:
```
```   590   assumes a: "Quotient R Abs Rep"
```
```   591   shows "(R ===> R ===> op =) R R"
```
```   592   apply(rule fun_relI)+
```
```   593   apply(rule equals_rsp[OF a])
```
```   594   apply(assumption)+
```
```   595   done
```
```   596
```
```   597 lemma o_prs:
```
```   598   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   599   and     q2: "Quotient R2 Abs2 Rep2"
```
```   600   and     q3: "Quotient R3 Abs3 Rep3"
```
```   601   shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
```
```   602   and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
```
```   603   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3]
```
```   604   by (simp_all add: fun_eq_iff)
```
```   605
```
```   606 lemma o_rsp:
```
```   607   "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
```
```   608   "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
```
```   609   by (force elim: fun_relE)+
```
```   610
```
```   611 lemma cond_prs:
```
```   612   assumes a: "Quotient R absf repf"
```
```   613   shows "absf (if a then repf b else repf c) = (if a then b else c)"
```
```   614   using a unfolding Quotient_def by auto
```
```   615
```
```   616 lemma if_prs:
```
```   617   assumes q: "Quotient R Abs Rep"
```
```   618   shows "(id ---> Rep ---> Rep ---> Abs) If = If"
```
```   619   using Quotient_abs_rep[OF q]
```
```   620   by (auto simp add: fun_eq_iff)
```
```   621
```
```   622 lemma if_rsp:
```
```   623   assumes q: "Quotient R Abs Rep"
```
```   624   shows "(op = ===> R ===> R ===> R) If If"
```
```   625   by force
```
```   626
```
```   627 lemma let_prs:
```
```   628   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   629   and     q2: "Quotient R2 Abs2 Rep2"
```
```   630   shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
```
```   631   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
```
```   632   by (auto simp add: fun_eq_iff)
```
```   633
```
```   634 lemma let_rsp:
```
```   635   shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
```
```   636   by (force elim: fun_relE)
```
```   637
```
```   638 lemma id_rsp:
```
```   639   shows "(R ===> R) id id"
```
```   640   by auto
```
```   641
```
```   642 lemma id_prs:
```
```   643   assumes a: "Quotient R Abs Rep"
```
```   644   shows "(Rep ---> Abs) id = id"
```
```   645   by (simp add: fun_eq_iff Quotient_abs_rep [OF a])
```
```   646
```
```   647
```
```   648 locale quot_type =
```
```   649   fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   650   and   Abs :: "'a set \<Rightarrow> 'b"
```
```   651   and   Rep :: "'b \<Rightarrow> 'a set"
```
```   652   assumes equivp: "part_equivp R"
```
```   653   and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = Collect (R x)"
```
```   654   and     rep_inverse: "\<And>x. Abs (Rep x) = x"
```
```   655   and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = Collect (R x)))) \<Longrightarrow> (Rep (Abs c)) = c"
```
```   656   and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
```
```   657 begin
```
```   658
```
```   659 definition
```
```   660   abs :: "'a \<Rightarrow> 'b"
```
```   661 where
```
```   662   "abs x = Abs (Collect (R x))"
```
```   663
```
```   664 definition
```
```   665   rep :: "'b \<Rightarrow> 'a"
```
```   666 where
```
```   667   "rep a = (SOME x. x \<in> Rep a)"
```
```   668
```
```   669 lemma some_collect:
```
```   670   assumes "R r r"
```
```   671   shows "R (SOME x. x \<in> Collect (R r)) = R r"
```
```   672   apply simp
```
```   673   by (metis assms exE_some equivp[simplified part_equivp_def])
```
```   674
```
```   675 lemma Quotient:
```
```   676   shows "Quotient R abs rep"
```
```   677   unfolding Quotient_def abs_def rep_def
```
```   678   proof (intro conjI allI)
```
```   679     fix a r s
```
```   680     show x: "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)" proof -
```
```   681       obtain x where r: "R x x" and rep: "Rep a = Collect (R x)" using rep_prop[of a] by auto
```
```   682       have "R (SOME x. x \<in> Rep a) x"  using r rep some_collect by metis
```
```   683       then have "R x (SOME x. x \<in> Rep a)" using part_equivp_symp[OF equivp] by fast
```
```   684       then show "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)"
```
```   685         using part_equivp_transp[OF equivp] by (metis `R (SOME x. x \<in> Rep a) x`)
```
```   686     qed
```
```   687     have "Collect (R (SOME x. x \<in> Rep a)) = (Rep a)" by (metis some_collect rep_prop)
```
```   688     then show "Abs (Collect (R (SOME x. x \<in> Rep a))) = a" using rep_inverse by auto
```
```   689     have "R r r \<Longrightarrow> R s s \<Longrightarrow> Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s"
```
```   690     proof -
```
```   691       assume "R r r" and "R s s"
```
```   692       then have "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> Collect (R r) = Collect (R s)"
```
```   693         by (metis abs_inverse)
```
```   694       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))"
```
```   695         by rule simp_all
```
```   696       finally show "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s" by simp
```
```   697     qed
```
```   698     then show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (Collect (R r)) = Abs (Collect (R s)))"
```
```   699       using equivp[simplified part_equivp_def] by metis
```
```   700     qed
```
```   701
```
```   702 end
```
```   703
```
```   704 subsection {* Quotient composition *}
```
```   705
```
```   706 lemma OOO_quotient:
```
```   707   fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   708   fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
```
```   709   fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
```
```   710   fixes R2' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   711   fixes R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"
```
```   712   assumes R1: "Quotient R1 Abs1 Rep1"
```
```   713   assumes R2: "Quotient R2 Abs2 Rep2"
```
```   714   assumes Abs1: "\<And>x y. R2' x y \<Longrightarrow> R1 x x \<Longrightarrow> R1 y y \<Longrightarrow> R2 (Abs1 x) (Abs1 y)"
```
```   715   assumes Rep1: "\<And>x y. R2 x y \<Longrightarrow> R2' (Rep1 x) (Rep1 y)"
```
```   716   shows "Quotient (R1 OO R2' OO R1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
```
```   717 apply (rule QuotientI)
```
```   718    apply (simp add: o_def Quotient_abs_rep [OF R2] Quotient_abs_rep [OF R1])
```
```   719   apply simp
```
```   720   apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI)
```
```   721    apply (rule Quotient_rep_reflp [OF R1])
```
```   722   apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI [rotated])
```
```   723    apply (rule Quotient_rep_reflp [OF R1])
```
```   724   apply (rule Rep1)
```
```   725   apply (rule Quotient_rep_reflp [OF R2])
```
```   726  apply safe
```
```   727     apply (rename_tac x y)
```
```   728     apply (drule Abs1)
```
```   729       apply (erule Quotient_refl2 [OF R1])
```
```   730      apply (erule Quotient_refl1 [OF R1])
```
```   731     apply (drule Quotient_refl1 [OF R2], drule Rep1)
```
```   732     apply (subgoal_tac "R1 r (Rep1 (Abs1 x))")
```
```   733      apply (rule_tac b="Rep1 (Abs1 x)" in relcomppI, assumption)
```
```   734      apply (erule relcomppI)
```
```   735      apply (erule Quotient_symp [OF R1, THEN sympD])
```
```   736     apply (rule Quotient_rel[symmetric, OF R1, THEN iffD2])
```
```   737     apply (rule conjI, erule Quotient_refl1 [OF R1])
```
```   738     apply (rule conjI, rule Quotient_rep_reflp [OF R1])
```
```   739     apply (subst Quotient_abs_rep [OF R1])
```
```   740     apply (erule Quotient_rel_abs [OF R1])
```
```   741    apply (rename_tac x y)
```
```   742    apply (drule Abs1)
```
```   743      apply (erule Quotient_refl2 [OF R1])
```
```   744     apply (erule Quotient_refl1 [OF R1])
```
```   745    apply (drule Quotient_refl2 [OF R2], drule Rep1)
```
```   746    apply (subgoal_tac "R1 s (Rep1 (Abs1 y))")
```
```   747     apply (rule_tac b="Rep1 (Abs1 y)" in relcomppI, assumption)
```
```   748     apply (erule relcomppI)
```
```   749     apply (erule Quotient_symp [OF R1, THEN sympD])
```
```   750    apply (rule Quotient_rel[symmetric, OF R1, THEN iffD2])
```
```   751    apply (rule conjI, erule Quotient_refl2 [OF R1])
```
```   752    apply (rule conjI, rule Quotient_rep_reflp [OF R1])
```
```   753    apply (subst Quotient_abs_rep [OF R1])
```
```   754    apply (erule Quotient_rel_abs [OF R1, THEN sym])
```
```   755   apply simp
```
```   756   apply (rule Quotient_rel_abs [OF R2])
```
```   757   apply (rule Quotient_rel_abs [OF R1, THEN ssubst], assumption)
```
```   758   apply (rule Quotient_rel_abs [OF R1, THEN subst], assumption)
```
```   759   apply (erule Abs1)
```
```   760    apply (erule Quotient_refl2 [OF R1])
```
```   761   apply (erule Quotient_refl1 [OF R1])
```
```   762  apply (rename_tac a b c d)
```
```   763  apply simp
```
```   764  apply (rule_tac b="Rep1 (Abs1 r)" in relcomppI)
```
```   765   apply (rule Quotient_rel[symmetric, OF R1, THEN iffD2])
```
```   766   apply (rule conjI, erule Quotient_refl1 [OF R1])
```
```   767   apply (simp add: Quotient_abs_rep [OF R1] Quotient_rep_reflp [OF R1])
```
```   768  apply (rule_tac b="Rep1 (Abs1 s)" in relcomppI [rotated])
```
```   769   apply (rule Quotient_rel[symmetric, OF R1, THEN iffD2])
```
```   770   apply (simp add: Quotient_abs_rep [OF R1] Quotient_rep_reflp [OF R1])
```
```   771   apply (erule Quotient_refl2 [OF R1])
```
```   772  apply (rule Rep1)
```
```   773  apply (drule Abs1)
```
```   774    apply (erule Quotient_refl2 [OF R1])
```
```   775   apply (erule Quotient_refl1 [OF R1])
```
```   776  apply (drule Abs1)
```
```   777   apply (erule Quotient_refl2 [OF R1])
```
```   778  apply (erule Quotient_refl1 [OF R1])
```
```   779  apply (drule Quotient_rel_abs [OF R1])
```
```   780  apply (drule Quotient_rel_abs [OF R1])
```
```   781  apply (drule Quotient_rel_abs [OF R1])
```
```   782  apply (drule Quotient_rel_abs [OF R1])
```
```   783  apply simp
```
```   784  apply (rule Quotient_rel[symmetric, OF R2, THEN iffD2])
```
```   785  apply simp
```
```   786 done
```
```   787
```
```   788 lemma OOO_eq_quotient:
```
```   789   fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   790   fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
```
```   791   fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
```
```   792   assumes R1: "Quotient R1 Abs1 Rep1"
```
```   793   assumes R2: "Quotient op= Abs2 Rep2"
```
```   794   shows "Quotient (R1 OOO op=) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
```
```   795 using assms
```
```   796 by (rule OOO_quotient) auto
```
```   797
```
```   798 subsection {* Invariant *}
```
```   799
```
```   800 definition invariant :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   801   where "invariant R = (\<lambda>x y. R x \<and> x = y)"
```
```   802
```
```   803 lemma invariant_to_eq:
```
```   804   assumes "invariant P x y"
```
```   805   shows "x = y"
```
```   806 using assms by (simp add: invariant_def)
```
```   807
```
```   808 lemma fun_rel_eq_invariant:
```
```   809   shows "((invariant R) ===> S) = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
```
```   810 by (auto simp add: invariant_def fun_rel_def)
```
```   811
```
```   812 lemma invariant_same_args:
```
```   813   shows "invariant P x x \<equiv> P x"
```
```   814 using assms by (auto simp add: invariant_def)
```
```   815
```
```   816 lemma copy_type_to_Quotient:
```
```   817   assumes "type_definition Rep Abs UNIV"
```
```   818   shows "Quotient (op =) Abs Rep"
```
```   819 proof -
```
```   820   interpret type_definition Rep Abs UNIV by fact
```
```   821   from Abs_inject Rep_inverse show ?thesis by (auto intro!: QuotientI)
```
```   822 qed
```
```   823
```
```   824 lemma copy_type_to_equivp:
```
```   825   fixes Abs :: "'a \<Rightarrow> 'b"
```
```   826   and Rep :: "'b \<Rightarrow> 'a"
```
```   827   assumes "type_definition Rep Abs (UNIV::'a set)"
```
```   828   shows "equivp (op=::'a\<Rightarrow>'a\<Rightarrow>bool)"
```
```   829 by (rule identity_equivp)
```
```   830
```
```   831 lemma invariant_type_to_Quotient:
```
```   832   assumes "type_definition Rep Abs {x. P x}"
```
```   833   shows "Quotient (invariant P) Abs Rep"
```
```   834 proof -
```
```   835   interpret type_definition Rep Abs "{x. P x}" by fact
```
```   836   from Rep Abs_inject Rep_inverse show ?thesis by (auto intro!: QuotientI simp: invariant_def)
```
```   837 qed
```
```   838
```
```   839 lemma invariant_type_to_part_equivp:
```
```   840   assumes "type_definition Rep Abs {x. P x}"
```
```   841   shows "part_equivp (invariant P)"
```
```   842 proof (intro part_equivpI)
```
```   843   interpret type_definition Rep Abs "{x. P x}" by fact
```
```   844   show "\<exists>x. invariant P x x" using Rep by (auto simp: invariant_def)
```
```   845 next
```
```   846   show "symp (invariant P)" by (auto intro: sympI simp: invariant_def)
```
```   847 next
```
```   848   show "transp (invariant P)" by (auto intro: transpI simp: invariant_def)
```
```   849 qed
```
```   850
```
```   851 subsection {* ML setup *}
```
```   852
```
```   853 text {* Auxiliary data for the quotient package *}
```
```   854
```
```   855 use "Tools/Quotient/quotient_info.ML"
```
```   856 setup Quotient_Info.setup
```
```   857
```
```   858 declare [[map "fun" = (fun_rel, fun_quotient)]]
```
```   859
```
```   860 lemmas [quot_thm] = fun_quotient
```
```   861 lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp id_rsp
```
```   862 lemmas [quot_preserve] = if_prs o_prs let_prs id_prs
```
```   863 lemmas [quot_equiv] = identity_equivp
```
```   864
```
```   865
```
```   866 text {* Lemmas about simplifying id's. *}
```
```   867 lemmas [id_simps] =
```
```   868   id_def[symmetric]
```
```   869   map_fun_id
```
```   870   id_apply
```
```   871   id_o
```
```   872   o_id
```
```   873   eq_comp_r
```
```   874   set_rel_eq
```
```   875   vimage_id
```
```   876
```
```   877 text {* Translation functions for the lifting process. *}
```
```   878 use "Tools/Quotient/quotient_term.ML"
```
```   879
```
```   880
```
```   881 text {* Definitions of the quotient types. *}
```
```   882 use "Tools/Quotient/quotient_type.ML"
```
```   883
```
```   884
```
```   885 text {* Definitions for quotient constants. *}
```
```   886 use "Tools/Quotient/quotient_def.ML"
```
```   887
```
```   888
```
```   889 text {*
```
```   890   An auxiliary constant for recording some information
```
```   891   about the lifted theorem in a tactic.
```
```   892 *}
```
```   893 definition
```
```   894   Quot_True :: "'a \<Rightarrow> bool"
```
```   895 where
```
```   896   "Quot_True x \<longleftrightarrow> True"
```
```   897
```
```   898 lemma
```
```   899   shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
```
```   900   and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
```
```   901   and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
```
```   902   and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
```
```   903   and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
```
```   904   by (simp_all add: Quot_True_def ext)
```
```   905
```
```   906 lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
```
```   907   by (simp add: Quot_True_def)
```
```   908
```
```   909
```
```   910 text {* Tactics for proving the lifted theorems *}
```
```   911 use "Tools/Quotient/quotient_tacs.ML"
```
```   912
```
```   913 subsection {* Methods / Interface *}
```
```   914
```
```   915 method_setup lifting =
```
```   916   {* Attrib.thms >> (fn thms => fn ctxt =>
```
```   917        SIMPLE_METHOD' (Quotient_Tacs.lift_tac ctxt [] thms)) *}
```
```   918   {* lift theorems to quotient types *}
```
```   919
```
```   920 method_setup lifting_setup =
```
```   921   {* Attrib.thm >> (fn thm => fn ctxt =>
```
```   922        SIMPLE_METHOD' (Quotient_Tacs.lift_procedure_tac ctxt [] thm)) *}
```
```   923   {* set up the three goals for the quotient lifting procedure *}
```
```   924
```
```   925 method_setup descending =
```
```   926   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_tac ctxt [])) *}
```
```   927   {* decend theorems to the raw level *}
```
```   928
```
```   929 method_setup descending_setup =
```
```   930   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_procedure_tac ctxt [])) *}
```
```   931   {* set up the three goals for the decending theorems *}
```
```   932
```
```   933 method_setup partiality_descending =
```
```   934   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_tac ctxt [])) *}
```
```   935   {* decend theorems to the raw level *}
```
```   936
```
```   937 method_setup partiality_descending_setup =
```
```   938   {* Scan.succeed (fn ctxt =>
```
```   939        SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_procedure_tac ctxt [])) *}
```
```   940   {* set up the three goals for the decending theorems *}
```
```   941
```
```   942 method_setup regularize =
```
```   943   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.regularize_tac ctxt)) *}
```
```   944   {* prove the regularization goals from the quotient lifting procedure *}
```
```   945
```
```   946 method_setup injection =
```
```   947   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.all_injection_tac ctxt)) *}
```
```   948   {* prove the rep/abs injection goals from the quotient lifting procedure *}
```
```   949
```
```   950 method_setup cleaning =
```
```   951   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.clean_tac ctxt)) *}
```
```   952   {* prove the cleaning goals from the quotient lifting procedure *}
```
```   953
```
```   954 attribute_setup quot_lifted =
```
```   955   {* Scan.succeed Quotient_Tacs.lifted_attrib *}
```
```   956   {* lift theorems to quotient types *}
```
```   957
```
```   958 no_notation
```
```   959   rel_conj (infixr "OOO" 75) and
```
```   960   map_fun (infixr "--->" 55) and
```
```   961   fun_rel (infixr "===>" 55)
```
```   962
```
```   963 hide_const (open) invariant
```
```   964
```
```   965 end
```