src/HOL/Quotient.thy
 author blanchet Mon Oct 04 22:45:09 2010 +0200 (2010-10-04) changeset 39946 78faa9b31202 parent 39669 9e3b035841e4 child 39956 132b79985660 permissions -rw-r--r--
move Metis into Plain
```     1 (*  Title:      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
```
```     9 uses
```
```    10   ("Tools/Quotient/quotient_info.ML")
```
```    11   ("Tools/Quotient/quotient_typ.ML")
```
```    12   ("Tools/Quotient/quotient_def.ML")
```
```    13   ("Tools/Quotient/quotient_term.ML")
```
```    14   ("Tools/Quotient/quotient_tacs.ML")
```
```    15 begin
```
```    16
```
```    17
```
```    18 text {*
```
```    19   Basic definition for equivalence relations
```
```    20   that are represented by predicates.
```
```    21 *}
```
```    22
```
```    23 definition
```
```    24   "equivp E \<equiv> \<forall>x y. E x y = (E x = E y)"
```
```    25
```
```    26 definition
```
```    27   "reflp E \<equiv> \<forall>x. E x x"
```
```    28
```
```    29 definition
```
```    30   "symp E \<equiv> \<forall>x y. E x y \<longrightarrow> E y x"
```
```    31
```
```    32 definition
```
```    33   "transp E \<equiv> \<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z"
```
```    34
```
```    35 lemma equivp_reflp_symp_transp:
```
```    36   shows "equivp E = (reflp E \<and> symp E \<and> transp E)"
```
```    37   unfolding equivp_def reflp_def symp_def transp_def fun_eq_iff
```
```    38   by blast
```
```    39
```
```    40 lemma equivp_reflp:
```
```    41   shows "equivp E \<Longrightarrow> E x x"
```
```    42   by (simp only: equivp_reflp_symp_transp reflp_def)
```
```    43
```
```    44 lemma equivp_symp:
```
```    45   shows "equivp E \<Longrightarrow> E x y \<Longrightarrow> E y x"
```
```    46   by (metis equivp_reflp_symp_transp symp_def)
```
```    47
```
```    48 lemma equivp_transp:
```
```    49   shows "equivp E \<Longrightarrow> E x y \<Longrightarrow> E y z \<Longrightarrow> E x z"
```
```    50   by (metis equivp_reflp_symp_transp transp_def)
```
```    51
```
```    52 lemma equivpI:
```
```    53   assumes "reflp R" "symp R" "transp R"
```
```    54   shows "equivp R"
```
```    55   using assms by (simp add: equivp_reflp_symp_transp)
```
```    56
```
```    57 lemma identity_equivp:
```
```    58   shows "equivp (op =)"
```
```    59   unfolding equivp_def
```
```    60   by auto
```
```    61
```
```    62 text {* Partial equivalences *}
```
```    63
```
```    64 definition
```
```    65   "part_equivp E \<equiv> (\<exists>x. E x x) \<and> (\<forall>x y. E x y = (E x x \<and> E y y \<and> (E x = E y)))"
```
```    66
```
```    67 lemma equivp_implies_part_equivp:
```
```    68   assumes a: "equivp E"
```
```    69   shows "part_equivp E"
```
```    70   using a
```
```    71   unfolding equivp_def part_equivp_def
```
```    72   by auto
```
```    73
```
```    74 lemma part_equivp_symp:
```
```    75   assumes e: "part_equivp R"
```
```    76   and a: "R x y"
```
```    77   shows "R y x"
```
```    78   using e[simplified part_equivp_def] a
```
```    79   by (metis)
```
```    80
```
```    81 lemma part_equivp_typedef:
```
```    82   shows "part_equivp R \<Longrightarrow> \<exists>d. d \<in> (\<lambda>c. \<exists>x. R x x \<and> c = R x)"
```
```    83   unfolding part_equivp_def mem_def
```
```    84   apply clarify
```
```    85   apply (intro exI)
```
```    86   apply (rule conjI)
```
```    87   apply assumption
```
```    88   apply (rule refl)
```
```    89   done
```
```    90
```
```    91 text {* Composition of Relations *}
```
```    92
```
```    93 abbreviation
```
```    94   rel_conj (infixr "OOO" 75)
```
```    95 where
```
```    96   "r1 OOO r2 \<equiv> r1 OO r2 OO r1"
```
```    97
```
```    98 lemma eq_comp_r:
```
```    99   shows "((op =) OOO R) = R"
```
```   100   by (auto simp add: fun_eq_iff)
```
```   101
```
```   102 subsection {* Respects predicate *}
```
```   103
```
```   104 definition
```
```   105   Respects
```
```   106 where
```
```   107   "Respects R x \<equiv> R x x"
```
```   108
```
```   109 lemma in_respects:
```
```   110   shows "(x \<in> Respects R) = R x x"
```
```   111   unfolding mem_def Respects_def
```
```   112   by simp
```
```   113
```
```   114 subsection {* Function map and function relation *}
```
```   115
```
```   116 definition
```
```   117   fun_map (infixr "--->" 55)
```
```   118 where
```
```   119 [simp]: "fun_map f g h x = g (h (f x))"
```
```   120
```
```   121 definition
```
```   122   fun_rel (infixr "===>" 55)
```
```   123 where
```
```   124 [simp]: "fun_rel E1 E2 f g = (\<forall>x y. E1 x y \<longrightarrow> E2 (f x) (g y))"
```
```   125
```
```   126 lemma fun_relI [intro]:
```
```   127   assumes "\<And>a b. P a b \<Longrightarrow> Q (x a) (y b)"
```
```   128   shows "(P ===> Q) x y"
```
```   129   using assms by (simp add: fun_rel_def)
```
```   130
```
```   131 lemma fun_map_id:
```
```   132   shows "(id ---> id) = id"
```
```   133   by (simp add: fun_eq_iff id_def)
```
```   134
```
```   135 lemma fun_rel_eq:
```
```   136   shows "((op =) ===> (op =)) = (op =)"
```
```   137   by (simp add: fun_eq_iff)
```
```   138
```
```   139
```
```   140 subsection {* Quotient Predicate *}
```
```   141
```
```   142 definition
```
```   143   "Quotient E Abs Rep \<equiv>
```
```   144      (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. E (Rep a) (Rep a)) \<and>
```
```   145      (\<forall>r s. E r s = (E r r \<and> E s s \<and> (Abs r = Abs s)))"
```
```   146
```
```   147 lemma Quotient_abs_rep:
```
```   148   assumes a: "Quotient E Abs Rep"
```
```   149   shows "Abs (Rep a) = a"
```
```   150   using a
```
```   151   unfolding Quotient_def
```
```   152   by simp
```
```   153
```
```   154 lemma Quotient_rep_reflp:
```
```   155   assumes a: "Quotient E Abs Rep"
```
```   156   shows "E (Rep a) (Rep a)"
```
```   157   using a
```
```   158   unfolding Quotient_def
```
```   159   by blast
```
```   160
```
```   161 lemma Quotient_rel:
```
```   162   assumes a: "Quotient E Abs Rep"
```
```   163   shows " E r s = (E r r \<and> E s s \<and> (Abs r = Abs s))"
```
```   164   using a
```
```   165   unfolding Quotient_def
```
```   166   by blast
```
```   167
```
```   168 lemma Quotient_rel_rep:
```
```   169   assumes a: "Quotient R Abs Rep"
```
```   170   shows "R (Rep a) (Rep b) = (a = b)"
```
```   171   using a
```
```   172   unfolding Quotient_def
```
```   173   by metis
```
```   174
```
```   175 lemma Quotient_rep_abs:
```
```   176   assumes a: "Quotient R Abs Rep"
```
```   177   shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
```
```   178   using a unfolding Quotient_def
```
```   179   by blast
```
```   180
```
```   181 lemma Quotient_rel_abs:
```
```   182   assumes a: "Quotient E Abs Rep"
```
```   183   shows "E r s \<Longrightarrow> Abs r = Abs s"
```
```   184   using a unfolding Quotient_def
```
```   185   by blast
```
```   186
```
```   187 lemma Quotient_symp:
```
```   188   assumes a: "Quotient E Abs Rep"
```
```   189   shows "symp E"
```
```   190   using a unfolding Quotient_def symp_def
```
```   191   by metis
```
```   192
```
```   193 lemma Quotient_transp:
```
```   194   assumes a: "Quotient E Abs Rep"
```
```   195   shows "transp E"
```
```   196   using a unfolding Quotient_def transp_def
```
```   197   by metis
```
```   198
```
```   199 lemma identity_quotient:
```
```   200   shows "Quotient (op =) id id"
```
```   201   unfolding Quotient_def id_def
```
```   202   by blast
```
```   203
```
```   204 lemma fun_quotient:
```
```   205   assumes q1: "Quotient R1 abs1 rep1"
```
```   206   and     q2: "Quotient R2 abs2 rep2"
```
```   207   shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
```
```   208 proof -
```
```   209   have "\<forall>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
```
```   210     using q1 q2
```
```   211     unfolding Quotient_def
```
```   212     unfolding fun_eq_iff
```
```   213     by simp
```
```   214   moreover
```
```   215   have "\<forall>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
```
```   216     using q1 q2
```
```   217     unfolding Quotient_def
```
```   218     by (simp (no_asm)) (metis)
```
```   219   moreover
```
```   220   have "\<forall>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
```
```   221         (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
```
```   222     unfolding fun_eq_iff
```
```   223     apply(auto)
```
```   224     using q1 q2 unfolding Quotient_def
```
```   225     apply(metis)
```
```   226     using q1 q2 unfolding Quotient_def
```
```   227     apply(metis)
```
```   228     using q1 q2 unfolding Quotient_def
```
```   229     apply(metis)
```
```   230     using q1 q2 unfolding Quotient_def
```
```   231     apply(metis)
```
```   232     done
```
```   233   ultimately
```
```   234   show "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
```
```   235     unfolding Quotient_def by blast
```
```   236 qed
```
```   237
```
```   238 lemma abs_o_rep:
```
```   239   assumes a: "Quotient R Abs Rep"
```
```   240   shows "Abs o Rep = id"
```
```   241   unfolding fun_eq_iff
```
```   242   by (simp add: Quotient_abs_rep[OF a])
```
```   243
```
```   244 lemma equals_rsp:
```
```   245   assumes q: "Quotient R Abs Rep"
```
```   246   and     a: "R xa xb" "R ya yb"
```
```   247   shows "R xa ya = R xb yb"
```
```   248   using a Quotient_symp[OF q] Quotient_transp[OF q]
```
```   249   unfolding symp_def transp_def
```
```   250   by blast
```
```   251
```
```   252 lemma lambda_prs:
```
```   253   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   254   and     q2: "Quotient R2 Abs2 Rep2"
```
```   255   shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
```
```   256   unfolding fun_eq_iff
```
```   257   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
```
```   258   by simp
```
```   259
```
```   260 lemma lambda_prs1:
```
```   261   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   262   and     q2: "Quotient R2 Abs2 Rep2"
```
```   263   shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
```
```   264   unfolding fun_eq_iff
```
```   265   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
```
```   266   by simp
```
```   267
```
```   268 lemma rep_abs_rsp:
```
```   269   assumes q: "Quotient R Abs Rep"
```
```   270   and     a: "R x1 x2"
```
```   271   shows "R x1 (Rep (Abs x2))"
```
```   272   using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
```
```   273   by metis
```
```   274
```
```   275 lemma rep_abs_rsp_left:
```
```   276   assumes q: "Quotient R Abs Rep"
```
```   277   and     a: "R x1 x2"
```
```   278   shows "R (Rep (Abs x1)) x2"
```
```   279   using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
```
```   280   by metis
```
```   281
```
```   282 text{*
```
```   283   In the following theorem R1 can be instantiated with anything,
```
```   284   but we know some of the types of the Rep and Abs functions;
```
```   285   so by solving Quotient assumptions we can get a unique R1 that
```
```   286   will be provable; which is why we need to use @{text apply_rsp} and
```
```   287   not the primed version *}
```
```   288
```
```   289 lemma apply_rsp:
```
```   290   fixes f g::"'a \<Rightarrow> 'c"
```
```   291   assumes q: "Quotient R1 Abs1 Rep1"
```
```   292   and     a: "(R1 ===> R2) f g" "R1 x y"
```
```   293   shows "R2 (f x) (g y)"
```
```   294   using a by simp
```
```   295
```
```   296 lemma apply_rsp':
```
```   297   assumes a: "(R1 ===> R2) f g" "R1 x y"
```
```   298   shows "R2 (f x) (g y)"
```
```   299   using a by simp
```
```   300
```
```   301 subsection {* lemmas for regularisation of ball and bex *}
```
```   302
```
```   303 lemma ball_reg_eqv:
```
```   304   fixes P :: "'a \<Rightarrow> bool"
```
```   305   assumes a: "equivp R"
```
```   306   shows "Ball (Respects R) P = (All P)"
```
```   307   using a
```
```   308   unfolding equivp_def
```
```   309   by (auto simp add: in_respects)
```
```   310
```
```   311 lemma bex_reg_eqv:
```
```   312   fixes P :: "'a \<Rightarrow> bool"
```
```   313   assumes a: "equivp R"
```
```   314   shows "Bex (Respects R) P = (Ex P)"
```
```   315   using a
```
```   316   unfolding equivp_def
```
```   317   by (auto simp add: in_respects)
```
```   318
```
```   319 lemma ball_reg_right:
```
```   320   assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x"
```
```   321   shows "All P \<longrightarrow> Ball R Q"
```
```   322   using a by (metis COMBC_def Collect_def Collect_mem_eq)
```
```   323
```
```   324 lemma bex_reg_left:
```
```   325   assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x"
```
```   326   shows "Bex R Q \<longrightarrow> Ex P"
```
```   327   using a by (metis COMBC_def Collect_def Collect_mem_eq)
```
```   328
```
```   329 lemma ball_reg_left:
```
```   330   assumes a: "equivp R"
```
```   331   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
```
```   332   using a by (metis equivp_reflp in_respects)
```
```   333
```
```   334 lemma bex_reg_right:
```
```   335   assumes a: "equivp R"
```
```   336   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
```
```   337   using a by (metis equivp_reflp in_respects)
```
```   338
```
```   339 lemma ball_reg_eqv_range:
```
```   340   fixes P::"'a \<Rightarrow> bool"
```
```   341   and x::"'a"
```
```   342   assumes a: "equivp R2"
```
```   343   shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
```
```   344   apply(rule iffI)
```
```   345   apply(rule allI)
```
```   346   apply(drule_tac x="\<lambda>y. f x" in bspec)
```
```   347   apply(simp add: in_respects)
```
```   348   apply(rule impI)
```
```   349   using a equivp_reflp_symp_transp[of "R2"]
```
```   350   apply(simp add: reflp_def)
```
```   351   apply(simp)
```
```   352   apply(simp)
```
```   353   done
```
```   354
```
```   355 lemma bex_reg_eqv_range:
```
```   356   assumes a: "equivp R2"
```
```   357   shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
```
```   358   apply(auto)
```
```   359   apply(rule_tac x="\<lambda>y. f x" in bexI)
```
```   360   apply(simp)
```
```   361   apply(simp add: Respects_def in_respects)
```
```   362   apply(rule impI)
```
```   363   using a equivp_reflp_symp_transp[of "R2"]
```
```   364   apply(simp add: reflp_def)
```
```   365   done
```
```   366
```
```   367 (* Next four lemmas are unused *)
```
```   368 lemma all_reg:
```
```   369   assumes a: "!x :: 'a. (P x --> Q x)"
```
```   370   and     b: "All P"
```
```   371   shows "All Q"
```
```   372   using a b by (metis)
```
```   373
```
```   374 lemma ex_reg:
```
```   375   assumes a: "!x :: 'a. (P x --> Q x)"
```
```   376   and     b: "Ex P"
```
```   377   shows "Ex Q"
```
```   378   using a b by metis
```
```   379
```
```   380 lemma ball_reg:
```
```   381   assumes a: "!x :: 'a. (R x --> P x --> Q x)"
```
```   382   and     b: "Ball R P"
```
```   383   shows "Ball R Q"
```
```   384   using a b by (metis COMBC_def Collect_def Collect_mem_eq)
```
```   385
```
```   386 lemma bex_reg:
```
```   387   assumes a: "!x :: 'a. (R x --> P x --> Q x)"
```
```   388   and     b: "Bex R P"
```
```   389   shows "Bex R Q"
```
```   390   using a b by (metis COMBC_def Collect_def Collect_mem_eq)
```
```   391
```
```   392
```
```   393 lemma ball_all_comm:
```
```   394   assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
```
```   395   shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
```
```   396   using assms by auto
```
```   397
```
```   398 lemma bex_ex_comm:
```
```   399   assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
```
```   400   shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
```
```   401   using assms by auto
```
```   402
```
```   403 subsection {* Bounded abstraction *}
```
```   404
```
```   405 definition
```
```   406   Babs :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
```
```   407 where
```
```   408   "x \<in> p \<Longrightarrow> Babs p m x = m x"
```
```   409
```
```   410 lemma babs_rsp:
```
```   411   assumes q: "Quotient R1 Abs1 Rep1"
```
```   412   and     a: "(R1 ===> R2) f g"
```
```   413   shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
```
```   414   apply (auto simp add: Babs_def in_respects)
```
```   415   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
```
```   416   using a apply (simp add: Babs_def)
```
```   417   apply (simp add: in_respects)
```
```   418   using Quotient_rel[OF q]
```
```   419   by metis
```
```   420
```
```   421 lemma babs_prs:
```
```   422   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   423   and     q2: "Quotient R2 Abs2 Rep2"
```
```   424   shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
```
```   425   apply (rule ext)
```
```   426   apply (simp)
```
```   427   apply (subgoal_tac "Rep1 x \<in> Respects R1")
```
```   428   apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
```
```   429   apply (simp add: in_respects Quotient_rel_rep[OF q1])
```
```   430   done
```
```   431
```
```   432 lemma babs_simp:
```
```   433   assumes q: "Quotient R1 Abs Rep"
```
```   434   shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
```
```   435   apply(rule iffI)
```
```   436   apply(simp_all only: babs_rsp[OF q])
```
```   437   apply(auto simp add: Babs_def)
```
```   438   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
```
```   439   apply(metis Babs_def)
```
```   440   apply (simp add: in_respects)
```
```   441   using Quotient_rel[OF q]
```
```   442   by metis
```
```   443
```
```   444 (* If a user proves that a particular functional relation
```
```   445    is an equivalence this may be useful in regularising *)
```
```   446 lemma babs_reg_eqv:
```
```   447   shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
```
```   448   by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
```
```   449
```
```   450
```
```   451 (* 3 lemmas needed for proving repabs_inj *)
```
```   452 lemma ball_rsp:
```
```   453   assumes a: "(R ===> (op =)) f g"
```
```   454   shows "Ball (Respects R) f = Ball (Respects R) g"
```
```   455   using a by (simp add: Ball_def in_respects)
```
```   456
```
```   457 lemma bex_rsp:
```
```   458   assumes a: "(R ===> (op =)) f g"
```
```   459   shows "(Bex (Respects R) f = Bex (Respects R) g)"
```
```   460   using a by (simp add: Bex_def in_respects)
```
```   461
```
```   462 lemma bex1_rsp:
```
```   463   assumes a: "(R ===> (op =)) f g"
```
```   464   shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
```
```   465   using a
```
```   466   by (simp add: Ex1_def in_respects) auto
```
```   467
```
```   468 (* 2 lemmas needed for cleaning of quantifiers *)
```
```   469 lemma all_prs:
```
```   470   assumes a: "Quotient R absf repf"
```
```   471   shows "Ball (Respects R) ((absf ---> id) f) = All f"
```
```   472   using a unfolding Quotient_def Ball_def in_respects fun_map_def id_apply
```
```   473   by metis
```
```   474
```
```   475 lemma ex_prs:
```
```   476   assumes a: "Quotient R absf repf"
```
```   477   shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
```
```   478   using a unfolding Quotient_def Bex_def in_respects fun_map_def id_apply
```
```   479   by metis
```
```   480
```
```   481 subsection {* @{text Bex1_rel} quantifier *}
```
```   482
```
```   483 definition
```
```   484   Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```   485 where
```
```   486   "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)))"
```
```   487
```
```   488 lemma bex1_rel_aux:
```
```   489   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
```
```   490   unfolding Bex1_rel_def
```
```   491   apply (erule conjE)+
```
```   492   apply (erule bexE)
```
```   493   apply rule
```
```   494   apply (rule_tac x="xa" in bexI)
```
```   495   apply metis
```
```   496   apply metis
```
```   497   apply rule+
```
```   498   apply (erule_tac x="xaa" in ballE)
```
```   499   prefer 2
```
```   500   apply (metis)
```
```   501   apply (erule_tac x="ya" in ballE)
```
```   502   prefer 2
```
```   503   apply (metis)
```
```   504   apply (metis in_respects)
```
```   505   done
```
```   506
```
```   507 lemma bex1_rel_aux2:
```
```   508   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
```
```   509   unfolding Bex1_rel_def
```
```   510   apply (erule conjE)+
```
```   511   apply (erule bexE)
```
```   512   apply rule
```
```   513   apply (rule_tac x="xa" in bexI)
```
```   514   apply metis
```
```   515   apply metis
```
```   516   apply rule+
```
```   517   apply (erule_tac x="xaa" in ballE)
```
```   518   prefer 2
```
```   519   apply (metis)
```
```   520   apply (erule_tac x="ya" in ballE)
```
```   521   prefer 2
```
```   522   apply (metis)
```
```   523   apply (metis in_respects)
```
```   524   done
```
```   525
```
```   526 lemma bex1_rel_rsp:
```
```   527   assumes a: "Quotient R absf repf"
```
```   528   shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
```
```   529   apply simp
```
```   530   apply clarify
```
```   531   apply rule
```
```   532   apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
```
```   533   apply (erule bex1_rel_aux2)
```
```   534   apply assumption
```
```   535   done
```
```   536
```
```   537
```
```   538 lemma ex1_prs:
```
```   539   assumes a: "Quotient R absf repf"
```
```   540   shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
```
```   541 apply simp
```
```   542 apply (subst Bex1_rel_def)
```
```   543 apply (subst Bex_def)
```
```   544 apply (subst Ex1_def)
```
```   545 apply simp
```
```   546 apply rule
```
```   547  apply (erule conjE)+
```
```   548  apply (erule_tac exE)
```
```   549  apply (erule conjE)
```
```   550  apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
```
```   551   apply (rule_tac x="absf x" in exI)
```
```   552   apply (simp)
```
```   553   apply rule+
```
```   554   using a unfolding Quotient_def
```
```   555   apply metis
```
```   556  apply rule+
```
```   557  apply (erule_tac x="x" in ballE)
```
```   558   apply (erule_tac x="y" in ballE)
```
```   559    apply simp
```
```   560   apply (simp add: in_respects)
```
```   561  apply (simp add: in_respects)
```
```   562 apply (erule_tac exE)
```
```   563  apply rule
```
```   564  apply (rule_tac x="repf x" in exI)
```
```   565  apply (simp only: in_respects)
```
```   566   apply rule
```
```   567  apply (metis Quotient_rel_rep[OF a])
```
```   568 using a unfolding Quotient_def apply (simp)
```
```   569 apply rule+
```
```   570 using a unfolding Quotient_def in_respects
```
```   571 apply metis
```
```   572 done
```
```   573
```
```   574 lemma bex1_bexeq_reg:
```
```   575   shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
```
```   576   apply (simp add: Ex1_def Bex1_rel_def in_respects)
```
```   577   apply clarify
```
```   578   apply auto
```
```   579   apply (rule bexI)
```
```   580   apply assumption
```
```   581   apply (simp add: in_respects)
```
```   582   apply (simp add: in_respects)
```
```   583   apply auto
```
```   584   done
```
```   585
```
```   586 lemma bex1_bexeq_reg_eqv:
```
```   587   assumes a: "equivp R"
```
```   588   shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
```
```   589   using equivp_reflp[OF a]
```
```   590   apply (intro impI)
```
```   591   apply (elim ex1E)
```
```   592   apply (rule mp[OF bex1_bexeq_reg])
```
```   593   apply (rule_tac a="x" in ex1I)
```
```   594   apply (subst in_respects)
```
```   595   apply (rule conjI)
```
```   596   apply assumption
```
```   597   apply assumption
```
```   598   apply clarify
```
```   599   apply (erule_tac x="xa" in allE)
```
```   600   apply simp
```
```   601   done
```
```   602
```
```   603 subsection {* Various respects and preserve lemmas *}
```
```   604
```
```   605 lemma quot_rel_rsp:
```
```   606   assumes a: "Quotient R Abs Rep"
```
```   607   shows "(R ===> R ===> op =) R R"
```
```   608   apply(rule fun_relI)+
```
```   609   apply(rule equals_rsp[OF a])
```
```   610   apply(assumption)+
```
```   611   done
```
```   612
```
```   613 lemma o_prs:
```
```   614   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   615   and     q2: "Quotient R2 Abs2 Rep2"
```
```   616   and     q3: "Quotient R3 Abs3 Rep3"
```
```   617   shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
```
```   618   and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
```
```   619   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3]
```
```   620   unfolding o_def fun_eq_iff by simp_all
```
```   621
```
```   622 lemma o_rsp:
```
```   623   "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
```
```   624   "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
```
```   625   unfolding fun_rel_def o_def fun_eq_iff by auto
```
```   626
```
```   627 lemma cond_prs:
```
```   628   assumes a: "Quotient R absf repf"
```
```   629   shows "absf (if a then repf b else repf c) = (if a then b else c)"
```
```   630   using a unfolding Quotient_def by auto
```
```   631
```
```   632 lemma if_prs:
```
```   633   assumes q: "Quotient R Abs Rep"
```
```   634   shows "(id ---> Rep ---> Rep ---> Abs) If = If"
```
```   635   using Quotient_abs_rep[OF q]
```
```   636   by (auto simp add: fun_eq_iff)
```
```   637
```
```   638 lemma if_rsp:
```
```   639   assumes q: "Quotient R Abs Rep"
```
```   640   shows "(op = ===> R ===> R ===> R) If If"
```
```   641   by auto
```
```   642
```
```   643 lemma let_prs:
```
```   644   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   645   and     q2: "Quotient R2 Abs2 Rep2"
```
```   646   shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
```
```   647   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
```
```   648   by (auto simp add: fun_eq_iff)
```
```   649
```
```   650 lemma let_rsp:
```
```   651   shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
```
```   652   by auto
```
```   653
```
```   654 lemma mem_rsp:
```
```   655   shows "(R1 ===> (R1 ===> R2) ===> R2) op \<in> op \<in>"
```
```   656   by (simp add: mem_def)
```
```   657
```
```   658 lemma mem_prs:
```
```   659   assumes a1: "Quotient R1 Abs1 Rep1"
```
```   660   and     a2: "Quotient R2 Abs2 Rep2"
```
```   661   shows "(Rep1 ---> (Abs1 ---> Rep2) ---> Abs2) op \<in> = op \<in>"
```
```   662   by (simp add: fun_eq_iff mem_def Quotient_abs_rep[OF a1] Quotient_abs_rep[OF a2])
```
```   663
```
```   664 lemma id_rsp:
```
```   665   shows "(R ===> R) id id"
```
```   666   by simp
```
```   667
```
```   668 lemma id_prs:
```
```   669   assumes a: "Quotient R Abs Rep"
```
```   670   shows "(Rep ---> Abs) id = id"
```
```   671   unfolding fun_eq_iff
```
```   672   by (simp add: Quotient_abs_rep[OF a])
```
```   673
```
```   674
```
```   675 locale quot_type =
```
```   676   fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   677   and   Abs :: "('a \<Rightarrow> bool) \<Rightarrow> 'b"
```
```   678   and   Rep :: "'b \<Rightarrow> ('a \<Rightarrow> bool)"
```
```   679   assumes equivp: "part_equivp R"
```
```   680   and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = R x"
```
```   681   and     rep_inverse: "\<And>x. Abs (Rep x) = x"
```
```   682   and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = R x))) \<Longrightarrow> (Rep (Abs c)) = c"
```
```   683   and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
```
```   684 begin
```
```   685
```
```   686 definition
```
```   687   abs::"'a \<Rightarrow> 'b"
```
```   688 where
```
```   689   "abs x \<equiv> Abs (R x)"
```
```   690
```
```   691 definition
```
```   692   rep::"'b \<Rightarrow> 'a"
```
```   693 where
```
```   694   "rep a = Eps (Rep a)"
```
```   695
```
```   696 lemma homeier5:
```
```   697   assumes a: "R r r"
```
```   698   shows "Rep (Abs (R r)) = R r"
```
```   699   apply (subst abs_inverse)
```
```   700   using a by auto
```
```   701
```
```   702 theorem homeier6:
```
```   703   assumes a: "R r r"
```
```   704   and b: "R s s"
```
```   705   shows "Abs (R r) = Abs (R s) \<longleftrightarrow> R r = R s"
```
```   706   by (metis a b homeier5)
```
```   707
```
```   708 theorem homeier8:
```
```   709   assumes "R r r"
```
```   710   shows "R (Eps (R r)) = R r"
```
```   711   using assms equivp[simplified part_equivp_def]
```
```   712   apply clarify
```
```   713   by (metis assms exE_some)
```
```   714
```
```   715 lemma Quotient:
```
```   716   shows "Quotient R abs rep"
```
```   717   unfolding Quotient_def abs_def rep_def
```
```   718   proof (intro conjI allI)
```
```   719     fix a r s
```
```   720     show "Abs (R (Eps (Rep a))) = a"
```
```   721       by (metis equivp exE_some part_equivp_def rep_inverse rep_prop)
```
```   722     show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (R r) = Abs (R s))"
```
```   723       by (metis homeier6 equivp[simplified part_equivp_def])
```
```   724     show "R (Eps (Rep a)) (Eps (Rep a))" proof -
```
```   725       obtain x where r: "R x x" and rep: "Rep a = R x" using rep_prop[of a] by auto
```
```   726       have "R (Eps (R x)) x" using homeier8 r by simp
```
```   727       then have "R x (Eps (R x))" using part_equivp_symp[OF equivp] by fast
```
```   728       then have "R (Eps (R x)) (Eps (R x))" using homeier8[OF r] by simp
```
```   729       then show "R (Eps (Rep a)) (Eps (Rep a))" using rep by simp
```
```   730     qed
```
```   731   qed
```
```   732
```
```   733 end
```
```   734
```
```   735
```
```   736 subsection {* ML setup *}
```
```   737
```
```   738 text {* Auxiliary data for the quotient package *}
```
```   739
```
```   740 use "Tools/Quotient/quotient_info.ML"
```
```   741
```
```   742 declare [[map "fun" = (fun_map, fun_rel)]]
```
```   743
```
```   744 lemmas [quot_thm] = fun_quotient
```
```   745 lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp mem_rsp id_rsp
```
```   746 lemmas [quot_preserve] = if_prs o_prs let_prs mem_prs id_prs
```
```   747 lemmas [quot_equiv] = identity_equivp
```
```   748
```
```   749
```
```   750 text {* Lemmas about simplifying id's. *}
```
```   751 lemmas [id_simps] =
```
```   752   id_def[symmetric]
```
```   753   fun_map_id
```
```   754   id_apply
```
```   755   id_o
```
```   756   o_id
```
```   757   eq_comp_r
```
```   758
```
```   759 text {* Translation functions for the lifting process. *}
```
```   760 use "Tools/Quotient/quotient_term.ML"
```
```   761
```
```   762
```
```   763 text {* Definitions of the quotient types. *}
```
```   764 use "Tools/Quotient/quotient_typ.ML"
```
```   765
```
```   766
```
```   767 text {* Definitions for quotient constants. *}
```
```   768 use "Tools/Quotient/quotient_def.ML"
```
```   769
```
```   770
```
```   771 text {*
```
```   772   An auxiliary constant for recording some information
```
```   773   about the lifted theorem in a tactic.
```
```   774 *}
```
```   775 definition
```
```   776   "Quot_True (x :: 'a) \<equiv> True"
```
```   777
```
```   778 lemma
```
```   779   shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
```
```   780   and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
```
```   781   and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
```
```   782   and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
```
```   783   and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
```
```   784   by (simp_all add: Quot_True_def ext)
```
```   785
```
```   786 lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
```
```   787   by (simp add: Quot_True_def)
```
```   788
```
```   789
```
```   790 text {* Tactics for proving the lifted theorems *}
```
```   791 use "Tools/Quotient/quotient_tacs.ML"
```
```   792
```
```   793 subsection {* Methods / Interface *}
```
```   794
```
```   795 method_setup lifting =
```
```   796   {* Attrib.thms >> (fn thms => fn ctxt =>
```
```   797        SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_tac ctxt [] thms))) *}
```
```   798   {* lifts theorems to quotient types *}
```
```   799
```
```   800 method_setup lifting_setup =
```
```   801   {* Attrib.thm >> (fn thm => fn ctxt =>
```
```   802        SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_procedure_tac ctxt [] thm))) *}
```
```   803   {* sets up the three goals for the quotient lifting procedure *}
```
```   804
```
```   805 method_setup descending =
```
```   806   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.descend_tac ctxt []))) *}
```
```   807   {* decends theorems to the raw level *}
```
```   808
```
```   809 method_setup descending_setup =
```
```   810   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.descend_procedure_tac ctxt []))) *}
```
```   811   {* sets up the three goals for the decending theorems *}
```
```   812
```
```   813 method_setup regularize =
```
```   814   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.regularize_tac ctxt))) *}
```
```   815   {* proves the regularization goals from the quotient lifting procedure *}
```
```   816
```
```   817 method_setup injection =
```
```   818   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.all_injection_tac ctxt))) *}
```
```   819   {* proves the rep/abs injection goals from the quotient lifting procedure *}
```
```   820
```
```   821 method_setup cleaning =
```
```   822   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.clean_tac ctxt))) *}
```
```   823   {* proves the cleaning goals from the quotient lifting procedure *}
```
```   824
```
```   825 attribute_setup quot_lifted =
```
```   826   {* Scan.succeed Quotient_Tacs.lifted_attrib *}
```
```   827   {* lifts theorems to quotient types *}
```
```   828
```
```   829 no_notation
```
```   830   rel_conj (infixr "OOO" 75) and
```
```   831   fun_map (infixr "--->" 55) and
```
```   832   fun_rel (infixr "===>" 55)
```
```   833
```
```   834 end
```
```   835
```