src/HOL/Quotient.thy
 author kuncar Fri Mar 23 14:03:58 2012 +0100 (2012-03-23) changeset 47091 d5cd13aca90b parent 46950 d0181abdbdac child 47094 1a7ad2601cb5 permissions -rw-r--r--
respectfulness theorem has to be proved if a new constant is lifted by quotient_definition
```     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   "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 {* Function map and function relation *}
```
```    56
```
```    57 notation map_fun (infixr "--->" 55)
```
```    58
```
```    59 lemma map_fun_id:
```
```    60   "(id ---> id) = id"
```
```    61   by (simp add: fun_eq_iff)
```
```    62
```
```    63 definition
```
```    64   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)
```
```    65 where
```
```    66   "fun_rel R1 R2 = (\<lambda>f g. \<forall>x y. R1 x y \<longrightarrow> R2 (f x) (g y))"
```
```    67
```
```    68 lemma fun_relI [intro]:
```
```    69   assumes "\<And>x y. R1 x y \<Longrightarrow> R2 (f x) (g y)"
```
```    70   shows "(R1 ===> R2) f g"
```
```    71   using assms by (simp add: fun_rel_def)
```
```    72
```
```    73 lemma fun_relE:
```
```    74   assumes "(R1 ===> R2) f g" and "R1 x y"
```
```    75   obtains "R2 (f x) (g y)"
```
```    76   using assms by (simp add: fun_rel_def)
```
```    77
```
```    78 lemma fun_rel_eq:
```
```    79   shows "((op =) ===> (op =)) = (op =)"
```
```    80   by (auto simp add: fun_eq_iff elim: fun_relE)
```
```    81
```
```    82 lemma fun_rel_eq_rel:
```
```    83   shows "((op =) ===> R) = (\<lambda>f g. \<forall>x. R (f x) (g x))"
```
```    84   by (simp add: fun_rel_def)
```
```    85
```
```    86 subsection {* set map (vimage) and set relation *}
```
```    87
```
```    88 definition "set_rel R xs ys \<equiv> \<forall>x y. R x y \<longrightarrow> x \<in> xs \<longleftrightarrow> y \<in> ys"
```
```    89
```
```    90 lemma vimage_id:
```
```    91   "vimage id = id"
```
```    92   unfolding vimage_def fun_eq_iff by auto
```
```    93
```
```    94 lemma set_rel_eq:
```
```    95   "set_rel op = = op ="
```
```    96   by (subst fun_eq_iff, subst fun_eq_iff) (simp add: set_eq_iff set_rel_def)
```
```    97
```
```    98 lemma set_rel_equivp:
```
```    99   assumes e: "equivp R"
```
```   100   shows "set_rel R xs ys \<longleftrightarrow> xs = ys \<and> (\<forall>x y. x \<in> xs \<longrightarrow> R x y \<longrightarrow> y \<in> xs)"
```
```   101   unfolding set_rel_def
```
```   102   using equivp_reflp[OF e]
```
```   103   by auto (metis, metis equivp_symp[OF e])
```
```   104
```
```   105 subsection {* Quotient Predicate *}
```
```   106
```
```   107 definition
```
```   108   "Quotient R Abs Rep \<longleftrightarrow>
```
```   109      (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and>
```
```   110      (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s)"
```
```   111
```
```   112 lemma QuotientI:
```
```   113   assumes "\<And>a. Abs (Rep a) = a"
```
```   114     and "\<And>a. R (Rep a) (Rep a)"
```
```   115     and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
```
```   116   shows "Quotient R Abs Rep"
```
```   117   using assms unfolding Quotient_def by blast
```
```   118
```
```   119 lemma Quotient_abs_rep:
```
```   120   assumes a: "Quotient R Abs Rep"
```
```   121   shows "Abs (Rep a) = a"
```
```   122   using a
```
```   123   unfolding Quotient_def
```
```   124   by simp
```
```   125
```
```   126 lemma Quotient_rep_reflp:
```
```   127   assumes a: "Quotient R Abs Rep"
```
```   128   shows "R (Rep a) (Rep a)"
```
```   129   using a
```
```   130   unfolding Quotient_def
```
```   131   by blast
```
```   132
```
```   133 lemma Quotient_rel:
```
```   134   assumes a: "Quotient R Abs Rep"
```
```   135   shows "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
```
```   136   using a
```
```   137   unfolding Quotient_def
```
```   138   by blast
```
```   139
```
```   140 lemma Quotient_rel_rep:
```
```   141   assumes a: "Quotient R Abs Rep"
```
```   142   shows "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
```
```   143   using a
```
```   144   unfolding Quotient_def
```
```   145   by metis
```
```   146
```
```   147 lemma Quotient_rep_abs:
```
```   148   assumes a: "Quotient R Abs Rep"
```
```   149   shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
```
```   150   using a unfolding Quotient_def
```
```   151   by blast
```
```   152
```
```   153 lemma Quotient_rel_abs:
```
```   154   assumes a: "Quotient R Abs Rep"
```
```   155   shows "R r s \<Longrightarrow> Abs r = Abs s"
```
```   156   using a unfolding Quotient_def
```
```   157   by blast
```
```   158
```
```   159 lemma Quotient_symp:
```
```   160   assumes a: "Quotient R Abs Rep"
```
```   161   shows "symp R"
```
```   162   using a unfolding Quotient_def using sympI by metis
```
```   163
```
```   164 lemma Quotient_transp:
```
```   165   assumes a: "Quotient R Abs Rep"
```
```   166   shows "transp R"
```
```   167   using a unfolding Quotient_def using transpI by metis
```
```   168
```
```   169 lemma identity_quotient:
```
```   170   shows "Quotient (op =) id id"
```
```   171   unfolding Quotient_def id_def
```
```   172   by blast
```
```   173
```
```   174 lemma fun_quotient:
```
```   175   assumes q1: "Quotient R1 abs1 rep1"
```
```   176   and     q2: "Quotient R2 abs2 rep2"
```
```   177   shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
```
```   178 proof -
```
```   179   have "\<And>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
```
```   180     using q1 q2 by (simp add: Quotient_def fun_eq_iff)
```
```   181   moreover
```
```   182   have "\<And>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
```
```   183     by (rule fun_relI)
```
```   184       (insert q1 q2 Quotient_rel_abs [of R1 abs1 rep1] Quotient_rel_rep [of R2 abs2 rep2],
```
```   185         simp (no_asm) add: Quotient_def, simp)
```
```   186   moreover
```
```   187   have "\<And>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
```
```   188         (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
```
```   189     apply(auto simp add: fun_rel_def fun_eq_iff)
```
```   190     using q1 q2 unfolding Quotient_def
```
```   191     apply(metis)
```
```   192     using q1 q2 unfolding Quotient_def
```
```   193     apply(metis)
```
```   194     using q1 q2 unfolding Quotient_def
```
```   195     apply(metis)
```
```   196     using q1 q2 unfolding Quotient_def
```
```   197     apply(metis)
```
```   198     done
```
```   199   ultimately
```
```   200   show "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
```
```   201     unfolding Quotient_def by blast
```
```   202 qed
```
```   203
```
```   204 lemma abs_o_rep:
```
```   205   assumes a: "Quotient R Abs Rep"
```
```   206   shows "Abs o Rep = id"
```
```   207   unfolding fun_eq_iff
```
```   208   by (simp add: Quotient_abs_rep[OF a])
```
```   209
```
```   210 lemma equals_rsp:
```
```   211   assumes q: "Quotient R Abs Rep"
```
```   212   and     a: "R xa xb" "R ya yb"
```
```   213   shows "R xa ya = R xb yb"
```
```   214   using a Quotient_symp[OF q] Quotient_transp[OF q]
```
```   215   by (blast elim: sympE transpE)
```
```   216
```
```   217 lemma lambda_prs:
```
```   218   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   219   and     q2: "Quotient R2 Abs2 Rep2"
```
```   220   shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
```
```   221   unfolding fun_eq_iff
```
```   222   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
```
```   223   by simp
```
```   224
```
```   225 lemma lambda_prs1:
```
```   226   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   227   and     q2: "Quotient R2 Abs2 Rep2"
```
```   228   shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
```
```   229   unfolding fun_eq_iff
```
```   230   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
```
```   231   by simp
```
```   232
```
```   233 lemma rep_abs_rsp:
```
```   234   assumes q: "Quotient R Abs Rep"
```
```   235   and     a: "R x1 x2"
```
```   236   shows "R x1 (Rep (Abs x2))"
```
```   237   using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
```
```   238   by metis
```
```   239
```
```   240 lemma rep_abs_rsp_left:
```
```   241   assumes q: "Quotient R Abs Rep"
```
```   242   and     a: "R x1 x2"
```
```   243   shows "R (Rep (Abs x1)) x2"
```
```   244   using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
```
```   245   by metis
```
```   246
```
```   247 text{*
```
```   248   In the following theorem R1 can be instantiated with anything,
```
```   249   but we know some of the types of the Rep and Abs functions;
```
```   250   so by solving Quotient assumptions we can get a unique R1 that
```
```   251   will be provable; which is why we need to use @{text apply_rsp} and
```
```   252   not the primed version *}
```
```   253
```
```   254 lemma apply_rsp:
```
```   255   fixes f g::"'a \<Rightarrow> 'c"
```
```   256   assumes q: "Quotient R1 Abs1 Rep1"
```
```   257   and     a: "(R1 ===> R2) f g" "R1 x y"
```
```   258   shows "R2 (f x) (g y)"
```
```   259   using a by (auto elim: fun_relE)
```
```   260
```
```   261 lemma apply_rsp':
```
```   262   assumes a: "(R1 ===> R2) f g" "R1 x y"
```
```   263   shows "R2 (f x) (g y)"
```
```   264   using a by (auto elim: fun_relE)
```
```   265
```
```   266 subsection {* lemmas for regularisation of ball and bex *}
```
```   267
```
```   268 lemma ball_reg_eqv:
```
```   269   fixes P :: "'a \<Rightarrow> bool"
```
```   270   assumes a: "equivp R"
```
```   271   shows "Ball (Respects R) P = (All P)"
```
```   272   using a
```
```   273   unfolding equivp_def
```
```   274   by (auto simp add: in_respects)
```
```   275
```
```   276 lemma bex_reg_eqv:
```
```   277   fixes P :: "'a \<Rightarrow> bool"
```
```   278   assumes a: "equivp R"
```
```   279   shows "Bex (Respects R) P = (Ex P)"
```
```   280   using a
```
```   281   unfolding equivp_def
```
```   282   by (auto simp add: in_respects)
```
```   283
```
```   284 lemma ball_reg_right:
```
```   285   assumes a: "\<And>x. x \<in> R \<Longrightarrow> P x \<longrightarrow> Q x"
```
```   286   shows "All P \<longrightarrow> Ball R Q"
```
```   287   using a by fast
```
```   288
```
```   289 lemma bex_reg_left:
```
```   290   assumes a: "\<And>x. x \<in> R \<Longrightarrow> Q x \<longrightarrow> P x"
```
```   291   shows "Bex R Q \<longrightarrow> Ex P"
```
```   292   using a by fast
```
```   293
```
```   294 lemma ball_reg_left:
```
```   295   assumes a: "equivp R"
```
```   296   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
```
```   297   using a by (metis equivp_reflp in_respects)
```
```   298
```
```   299 lemma bex_reg_right:
```
```   300   assumes a: "equivp R"
```
```   301   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
```
```   302   using a by (metis equivp_reflp in_respects)
```
```   303
```
```   304 lemma ball_reg_eqv_range:
```
```   305   fixes P::"'a \<Rightarrow> bool"
```
```   306   and x::"'a"
```
```   307   assumes a: "equivp R2"
```
```   308   shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
```
```   309   apply(rule iffI)
```
```   310   apply(rule allI)
```
```   311   apply(drule_tac x="\<lambda>y. f x" in bspec)
```
```   312   apply(simp add: in_respects fun_rel_def)
```
```   313   apply(rule impI)
```
```   314   using a equivp_reflp_symp_transp[of "R2"]
```
```   315   apply (auto elim: equivpE reflpE)
```
```   316   done
```
```   317
```
```   318 lemma bex_reg_eqv_range:
```
```   319   assumes a: "equivp R2"
```
```   320   shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
```
```   321   apply(auto)
```
```   322   apply(rule_tac x="\<lambda>y. f x" in bexI)
```
```   323   apply(simp)
```
```   324   apply(simp add: Respects_def in_respects fun_rel_def)
```
```   325   apply(rule impI)
```
```   326   using a equivp_reflp_symp_transp[of "R2"]
```
```   327   apply (auto elim: equivpE reflpE)
```
```   328   done
```
```   329
```
```   330 (* Next four lemmas are unused *)
```
```   331 lemma all_reg:
```
```   332   assumes a: "!x :: 'a. (P x --> Q x)"
```
```   333   and     b: "All P"
```
```   334   shows "All Q"
```
```   335   using a b by fast
```
```   336
```
```   337 lemma ex_reg:
```
```   338   assumes a: "!x :: 'a. (P x --> Q x)"
```
```   339   and     b: "Ex P"
```
```   340   shows "Ex Q"
```
```   341   using a b by fast
```
```   342
```
```   343 lemma ball_reg:
```
```   344   assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
```
```   345   and     b: "Ball R P"
```
```   346   shows "Ball R Q"
```
```   347   using a b by fast
```
```   348
```
```   349 lemma bex_reg:
```
```   350   assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
```
```   351   and     b: "Bex R P"
```
```   352   shows "Bex R Q"
```
```   353   using a b by fast
```
```   354
```
```   355
```
```   356 lemma ball_all_comm:
```
```   357   assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
```
```   358   shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
```
```   359   using assms by auto
```
```   360
```
```   361 lemma bex_ex_comm:
```
```   362   assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
```
```   363   shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
```
```   364   using assms by auto
```
```   365
```
```   366 subsection {* Bounded abstraction *}
```
```   367
```
```   368 definition
```
```   369   Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
```
```   370 where
```
```   371   "x \<in> p \<Longrightarrow> Babs p m x = m x"
```
```   372
```
```   373 lemma babs_rsp:
```
```   374   assumes q: "Quotient R1 Abs1 Rep1"
```
```   375   and     a: "(R1 ===> R2) f g"
```
```   376   shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
```
```   377   apply (auto simp add: Babs_def in_respects fun_rel_def)
```
```   378   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
```
```   379   using a apply (simp add: Babs_def fun_rel_def)
```
```   380   apply (simp add: in_respects fun_rel_def)
```
```   381   using Quotient_rel[OF q]
```
```   382   by metis
```
```   383
```
```   384 lemma babs_prs:
```
```   385   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   386   and     q2: "Quotient R2 Abs2 Rep2"
```
```   387   shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
```
```   388   apply (rule ext)
```
```   389   apply (simp add:)
```
```   390   apply (subgoal_tac "Rep1 x \<in> Respects R1")
```
```   391   apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
```
```   392   apply (simp add: in_respects Quotient_rel_rep[OF q1])
```
```   393   done
```
```   394
```
```   395 lemma babs_simp:
```
```   396   assumes q: "Quotient R1 Abs Rep"
```
```   397   shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
```
```   398   apply(rule iffI)
```
```   399   apply(simp_all only: babs_rsp[OF q])
```
```   400   apply(auto simp add: Babs_def fun_rel_def)
```
```   401   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
```
```   402   apply(metis Babs_def)
```
```   403   apply (simp add: in_respects)
```
```   404   using Quotient_rel[OF q]
```
```   405   by metis
```
```   406
```
```   407 (* If a user proves that a particular functional relation
```
```   408    is an equivalence this may be useful in regularising *)
```
```   409 lemma babs_reg_eqv:
```
```   410   shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
```
```   411   by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
```
```   412
```
```   413
```
```   414 (* 3 lemmas needed for proving repabs_inj *)
```
```   415 lemma ball_rsp:
```
```   416   assumes a: "(R ===> (op =)) f g"
```
```   417   shows "Ball (Respects R) f = Ball (Respects R) g"
```
```   418   using a by (auto simp add: Ball_def in_respects elim: fun_relE)
```
```   419
```
```   420 lemma bex_rsp:
```
```   421   assumes a: "(R ===> (op =)) f g"
```
```   422   shows "(Bex (Respects R) f = Bex (Respects R) g)"
```
```   423   using a by (auto simp add: Bex_def in_respects elim: fun_relE)
```
```   424
```
```   425 lemma bex1_rsp:
```
```   426   assumes a: "(R ===> (op =)) f g"
```
```   427   shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
```
```   428   using a by (auto elim: fun_relE simp add: Ex1_def in_respects)
```
```   429
```
```   430 (* 2 lemmas needed for cleaning of quantifiers *)
```
```   431 lemma all_prs:
```
```   432   assumes a: "Quotient R absf repf"
```
```   433   shows "Ball (Respects R) ((absf ---> id) f) = All f"
```
```   434   using a unfolding Quotient_def Ball_def in_respects id_apply comp_def map_fun_def
```
```   435   by metis
```
```   436
```
```   437 lemma ex_prs:
```
```   438   assumes a: "Quotient R absf repf"
```
```   439   shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
```
```   440   using a unfolding Quotient_def Bex_def in_respects id_apply comp_def map_fun_def
```
```   441   by metis
```
```   442
```
```   443 subsection {* @{text Bex1_rel} quantifier *}
```
```   444
```
```   445 definition
```
```   446   Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```   447 where
```
```   448   "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)))"
```
```   449
```
```   450 lemma bex1_rel_aux:
```
```   451   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
```
```   452   unfolding Bex1_rel_def
```
```   453   apply (erule conjE)+
```
```   454   apply (erule bexE)
```
```   455   apply rule
```
```   456   apply (rule_tac x="xa" in bexI)
```
```   457   apply metis
```
```   458   apply metis
```
```   459   apply rule+
```
```   460   apply (erule_tac x="xaa" in ballE)
```
```   461   prefer 2
```
```   462   apply (metis)
```
```   463   apply (erule_tac x="ya" in ballE)
```
```   464   prefer 2
```
```   465   apply (metis)
```
```   466   apply (metis in_respects)
```
```   467   done
```
```   468
```
```   469 lemma bex1_rel_aux2:
```
```   470   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
```
```   471   unfolding Bex1_rel_def
```
```   472   apply (erule conjE)+
```
```   473   apply (erule bexE)
```
```   474   apply rule
```
```   475   apply (rule_tac x="xa" in bexI)
```
```   476   apply metis
```
```   477   apply metis
```
```   478   apply rule+
```
```   479   apply (erule_tac x="xaa" in ballE)
```
```   480   prefer 2
```
```   481   apply (metis)
```
```   482   apply (erule_tac x="ya" in ballE)
```
```   483   prefer 2
```
```   484   apply (metis)
```
```   485   apply (metis in_respects)
```
```   486   done
```
```   487
```
```   488 lemma bex1_rel_rsp:
```
```   489   assumes a: "Quotient R absf repf"
```
```   490   shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
```
```   491   apply (simp add: fun_rel_def)
```
```   492   apply clarify
```
```   493   apply rule
```
```   494   apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
```
```   495   apply (erule bex1_rel_aux2)
```
```   496   apply assumption
```
```   497   done
```
```   498
```
```   499
```
```   500 lemma ex1_prs:
```
```   501   assumes a: "Quotient R absf repf"
```
```   502   shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
```
```   503 apply (simp add:)
```
```   504 apply (subst Bex1_rel_def)
```
```   505 apply (subst Bex_def)
```
```   506 apply (subst Ex1_def)
```
```   507 apply simp
```
```   508 apply rule
```
```   509  apply (erule conjE)+
```
```   510  apply (erule_tac exE)
```
```   511  apply (erule conjE)
```
```   512  apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
```
```   513   apply (rule_tac x="absf x" in exI)
```
```   514   apply (simp)
```
```   515   apply rule+
```
```   516   using a unfolding Quotient_def
```
```   517   apply metis
```
```   518  apply rule+
```
```   519  apply (erule_tac x="x" in ballE)
```
```   520   apply (erule_tac x="y" in ballE)
```
```   521    apply simp
```
```   522   apply (simp add: in_respects)
```
```   523  apply (simp add: in_respects)
```
```   524 apply (erule_tac exE)
```
```   525  apply rule
```
```   526  apply (rule_tac x="repf x" in exI)
```
```   527  apply (simp only: in_respects)
```
```   528   apply rule
```
```   529  apply (metis Quotient_rel_rep[OF a])
```
```   530 using a unfolding Quotient_def apply (simp)
```
```   531 apply rule+
```
```   532 using a unfolding Quotient_def in_respects
```
```   533 apply metis
```
```   534 done
```
```   535
```
```   536 lemma bex1_bexeq_reg:
```
```   537   shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
```
```   538   apply (simp add: Ex1_def Bex1_rel_def in_respects)
```
```   539   apply clarify
```
```   540   apply auto
```
```   541   apply (rule bexI)
```
```   542   apply assumption
```
```   543   apply (simp add: in_respects)
```
```   544   apply (simp add: in_respects)
```
```   545   apply auto
```
```   546   done
```
```   547
```
```   548 lemma bex1_bexeq_reg_eqv:
```
```   549   assumes a: "equivp R"
```
```   550   shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
```
```   551   using equivp_reflp[OF a]
```
```   552   apply (intro impI)
```
```   553   apply (elim ex1E)
```
```   554   apply (rule mp[OF bex1_bexeq_reg])
```
```   555   apply (rule_tac a="x" in ex1I)
```
```   556   apply (subst in_respects)
```
```   557   apply (rule conjI)
```
```   558   apply assumption
```
```   559   apply assumption
```
```   560   apply clarify
```
```   561   apply (erule_tac x="xa" in allE)
```
```   562   apply simp
```
```   563   done
```
```   564
```
```   565 subsection {* Various respects and preserve lemmas *}
```
```   566
```
```   567 lemma quot_rel_rsp:
```
```   568   assumes a: "Quotient R Abs Rep"
```
```   569   shows "(R ===> R ===> op =) R R"
```
```   570   apply(rule fun_relI)+
```
```   571   apply(rule equals_rsp[OF a])
```
```   572   apply(assumption)+
```
```   573   done
```
```   574
```
```   575 lemma o_prs:
```
```   576   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   577   and     q2: "Quotient R2 Abs2 Rep2"
```
```   578   and     q3: "Quotient R3 Abs3 Rep3"
```
```   579   shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
```
```   580   and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
```
```   581   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3]
```
```   582   by (simp_all add: fun_eq_iff)
```
```   583
```
```   584 lemma o_rsp:
```
```   585   "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
```
```   586   "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
```
```   587   by (force elim: fun_relE)+
```
```   588
```
```   589 lemma cond_prs:
```
```   590   assumes a: "Quotient R absf repf"
```
```   591   shows "absf (if a then repf b else repf c) = (if a then b else c)"
```
```   592   using a unfolding Quotient_def by auto
```
```   593
```
```   594 lemma if_prs:
```
```   595   assumes q: "Quotient R Abs Rep"
```
```   596   shows "(id ---> Rep ---> Rep ---> Abs) If = If"
```
```   597   using Quotient_abs_rep[OF q]
```
```   598   by (auto simp add: fun_eq_iff)
```
```   599
```
```   600 lemma if_rsp:
```
```   601   assumes q: "Quotient R Abs Rep"
```
```   602   shows "(op = ===> R ===> R ===> R) If If"
```
```   603   by force
```
```   604
```
```   605 lemma let_prs:
```
```   606   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   607   and     q2: "Quotient R2 Abs2 Rep2"
```
```   608   shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
```
```   609   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
```
```   610   by (auto simp add: fun_eq_iff)
```
```   611
```
```   612 lemma let_rsp:
```
```   613   shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
```
```   614   by (force elim: fun_relE)
```
```   615
```
```   616 lemma id_rsp:
```
```   617   shows "(R ===> R) id id"
```
```   618   by auto
```
```   619
```
```   620 lemma id_prs:
```
```   621   assumes a: "Quotient R Abs Rep"
```
```   622   shows "(Rep ---> Abs) id = id"
```
```   623   by (simp add: fun_eq_iff Quotient_abs_rep [OF a])
```
```   624
```
```   625
```
```   626 locale quot_type =
```
```   627   fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   628   and   Abs :: "'a set \<Rightarrow> 'b"
```
```   629   and   Rep :: "'b \<Rightarrow> 'a set"
```
```   630   assumes equivp: "part_equivp R"
```
```   631   and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = Collect (R x)"
```
```   632   and     rep_inverse: "\<And>x. Abs (Rep x) = x"
```
```   633   and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = Collect (R x)))) \<Longrightarrow> (Rep (Abs c)) = c"
```
```   634   and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
```
```   635 begin
```
```   636
```
```   637 definition
```
```   638   abs :: "'a \<Rightarrow> 'b"
```
```   639 where
```
```   640   "abs x = Abs (Collect (R x))"
```
```   641
```
```   642 definition
```
```   643   rep :: "'b \<Rightarrow> 'a"
```
```   644 where
```
```   645   "rep a = (SOME x. x \<in> Rep a)"
```
```   646
```
```   647 lemma some_collect:
```
```   648   assumes "R r r"
```
```   649   shows "R (SOME x. x \<in> Collect (R r)) = R r"
```
```   650   apply simp
```
```   651   by (metis assms exE_some equivp[simplified part_equivp_def])
```
```   652
```
```   653 lemma Quotient:
```
```   654   shows "Quotient R abs rep"
```
```   655   unfolding Quotient_def abs_def rep_def
```
```   656   proof (intro conjI allI)
```
```   657     fix a r s
```
```   658     show x: "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)" proof -
```
```   659       obtain x where r: "R x x" and rep: "Rep a = Collect (R x)" using rep_prop[of a] by auto
```
```   660       have "R (SOME x. x \<in> Rep a) x"  using r rep some_collect by metis
```
```   661       then have "R x (SOME x. x \<in> Rep a)" using part_equivp_symp[OF equivp] by fast
```
```   662       then show "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)"
```
```   663         using part_equivp_transp[OF equivp] by (metis `R (SOME x. x \<in> Rep a) x`)
```
```   664     qed
```
```   665     have "Collect (R (SOME x. x \<in> Rep a)) = (Rep a)" by (metis some_collect rep_prop)
```
```   666     then show "Abs (Collect (R (SOME x. x \<in> Rep a))) = a" using rep_inverse by auto
```
```   667     have "R r r \<Longrightarrow> R s s \<Longrightarrow> Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s"
```
```   668     proof -
```
```   669       assume "R r r" and "R s s"
```
```   670       then have "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> Collect (R r) = Collect (R s)"
```
```   671         by (metis abs_inverse)
```
```   672       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))"
```
```   673         by rule simp_all
```
```   674       finally show "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s" by simp
```
```   675     qed
```
```   676     then show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (Collect (R r)) = Abs (Collect (R s)))"
```
```   677       using equivp[simplified part_equivp_def] by metis
```
```   678     qed
```
```   679
```
```   680 end
```
```   681
```
```   682 subsection {* ML setup *}
```
```   683
```
```   684 text {* Auxiliary data for the quotient package *}
```
```   685
```
```   686 use "Tools/Quotient/quotient_info.ML"
```
```   687 setup Quotient_Info.setup
```
```   688
```
```   689 declare [[map "fun" = fun_rel]]
```
```   690 declare [[map set = set_rel]]
```
```   691
```
```   692 lemmas [quot_thm] = fun_quotient
```
```   693 lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp id_rsp
```
```   694 lemmas [quot_preserve] = if_prs o_prs let_prs id_prs
```
```   695 lemmas [quot_equiv] = identity_equivp
```
```   696
```
```   697
```
```   698 text {* Lemmas about simplifying id's. *}
```
```   699 lemmas [id_simps] =
```
```   700   id_def[symmetric]
```
```   701   map_fun_id
```
```   702   id_apply
```
```   703   id_o
```
```   704   o_id
```
```   705   eq_comp_r
```
```   706   set_rel_eq
```
```   707   vimage_id
```
```   708
```
```   709 text {* Translation functions for the lifting process. *}
```
```   710 use "Tools/Quotient/quotient_term.ML"
```
```   711
```
```   712
```
```   713 text {* Definitions of the quotient types. *}
```
```   714 use "Tools/Quotient/quotient_type.ML"
```
```   715
```
```   716
```
```   717 text {* Definitions for quotient constants. *}
```
```   718 use "Tools/Quotient/quotient_def.ML"
```
```   719
```
```   720
```
```   721 text {*
```
```   722   An auxiliary constant for recording some information
```
```   723   about the lifted theorem in a tactic.
```
```   724 *}
```
```   725 definition
```
```   726   Quot_True :: "'a \<Rightarrow> bool"
```
```   727 where
```
```   728   "Quot_True x \<longleftrightarrow> True"
```
```   729
```
```   730 lemma
```
```   731   shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
```
```   732   and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
```
```   733   and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
```
```   734   and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
```
```   735   and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
```
```   736   by (simp_all add: Quot_True_def ext)
```
```   737
```
```   738 lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
```
```   739   by (simp add: Quot_True_def)
```
```   740
```
```   741
```
```   742 text {* Tactics for proving the lifted theorems *}
```
```   743 use "Tools/Quotient/quotient_tacs.ML"
```
```   744
```
```   745 subsection {* Methods / Interface *}
```
```   746
```
```   747 method_setup lifting =
```
```   748   {* Attrib.thms >> (fn thms => fn ctxt =>
```
```   749        SIMPLE_METHOD' (Quotient_Tacs.lift_tac ctxt [] thms)) *}
```
```   750   {* lift theorems to quotient types *}
```
```   751
```
```   752 method_setup lifting_setup =
```
```   753   {* Attrib.thm >> (fn thm => fn ctxt =>
```
```   754        SIMPLE_METHOD' (Quotient_Tacs.lift_procedure_tac ctxt [] thm)) *}
```
```   755   {* set up the three goals for the quotient lifting procedure *}
```
```   756
```
```   757 method_setup descending =
```
```   758   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_tac ctxt [])) *}
```
```   759   {* decend theorems to the raw level *}
```
```   760
```
```   761 method_setup descending_setup =
```
```   762   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_procedure_tac ctxt [])) *}
```
```   763   {* set up the three goals for the decending theorems *}
```
```   764
```
```   765 method_setup partiality_descending =
```
```   766   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_tac ctxt [])) *}
```
```   767   {* decend theorems to the raw level *}
```
```   768
```
```   769 method_setup partiality_descending_setup =
```
```   770   {* Scan.succeed (fn ctxt =>
```
```   771        SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_procedure_tac ctxt [])) *}
```
```   772   {* set up the three goals for the decending theorems *}
```
```   773
```
```   774 method_setup regularize =
```
```   775   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.regularize_tac ctxt)) *}
```
```   776   {* prove the regularization goals from the quotient lifting procedure *}
```
```   777
```
```   778 method_setup injection =
```
```   779   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.all_injection_tac ctxt)) *}
```
```   780   {* prove the rep/abs injection goals from the quotient lifting procedure *}
```
```   781
```
```   782 method_setup cleaning =
```
```   783   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.clean_tac ctxt)) *}
```
```   784   {* prove the cleaning goals from the quotient lifting procedure *}
```
```   785
```
```   786 attribute_setup quot_lifted =
```
```   787   {* Scan.succeed Quotient_Tacs.lifted_attrib *}
```
```   788   {* lift theorems to quotient types *}
```
```   789
```
```   790 no_notation
```
```   791   rel_conj (infixr "OOO" 75) and
```
```   792   map_fun (infixr "--->" 55) and
```
```   793   fun_rel (infixr "===>" 55)
```
```   794
```
```   795 end
```