src/HOL/Quotient.thy
author wenzelm
Sun Mar 13 22:55:50 2011 +0100 (2011-03-13)
changeset 41959 b460124855b8
parent 41452 c291e0826902
child 42334 8e58cc1390c7
permissions -rw-r--r--
tuned headers;
     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 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 text {*
    18   Basic definition for equivalence relations
    19   that are represented by predicates.
    20 *}
    21 
    22 text {* Composition of Relations *}
    23 
    24 abbreviation
    25   rel_conj :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" (infixr "OOO" 75)
    26 where
    27   "r1 OOO r2 \<equiv> r1 OO r2 OO r1"
    28 
    29 lemma eq_comp_r:
    30   shows "((op =) OOO R) = R"
    31   by (auto simp add: fun_eq_iff)
    32 
    33 subsection {* Respects predicate *}
    34 
    35 definition
    36   Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
    37 where
    38   "Respects R x = R x x"
    39 
    40 lemma in_respects:
    41   shows "x \<in> Respects R \<longleftrightarrow> R x x"
    42   unfolding mem_def Respects_def
    43   by simp
    44 
    45 subsection {* Function map and function relation *}
    46 
    47 notation map_fun (infixr "--->" 55)
    48 
    49 lemma map_fun_id:
    50   "(id ---> id) = id"
    51   by (simp add: fun_eq_iff)
    52 
    53 definition
    54   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)
    55 where
    56   "fun_rel R1 R2 = (\<lambda>f g. \<forall>x y. R1 x y \<longrightarrow> R2 (f x) (g y))"
    57 
    58 lemma fun_relI [intro]:
    59   assumes "\<And>x y. R1 x y \<Longrightarrow> R2 (f x) (g y)"
    60   shows "(R1 ===> R2) f g"
    61   using assms by (simp add: fun_rel_def)
    62 
    63 lemma fun_relE:
    64   assumes "(R1 ===> R2) f g" and "R1 x y"
    65   obtains "R2 (f x) (g y)"
    66   using assms by (simp add: fun_rel_def)
    67 
    68 lemma fun_rel_eq:
    69   shows "((op =) ===> (op =)) = (op =)"
    70   by (auto simp add: fun_eq_iff elim: fun_relE)
    71 
    72 
    73 subsection {* Quotient Predicate *}
    74 
    75 definition
    76   "Quotient R Abs Rep \<longleftrightarrow>
    77      (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and>
    78      (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s)"
    79 
    80 lemma QuotientI:
    81   assumes "\<And>a. Abs (Rep a) = a"
    82     and "\<And>a. R (Rep a) (Rep a)"
    83     and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
    84   shows "Quotient R Abs Rep"
    85   using assms unfolding Quotient_def by blast
    86 
    87 lemma Quotient_abs_rep:
    88   assumes a: "Quotient R Abs Rep"
    89   shows "Abs (Rep a) = a"
    90   using a
    91   unfolding Quotient_def
    92   by simp
    93 
    94 lemma Quotient_rep_reflp:
    95   assumes a: "Quotient R Abs Rep"
    96   shows "R (Rep a) (Rep a)"
    97   using a
    98   unfolding Quotient_def
    99   by blast
   100 
   101 lemma Quotient_rel:
   102   assumes a: "Quotient R Abs Rep"
   103   shows "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
   104   using a
   105   unfolding Quotient_def
   106   by blast
   107 
   108 lemma Quotient_rel_rep:
   109   assumes a: "Quotient R Abs Rep"
   110   shows "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
   111   using a
   112   unfolding Quotient_def
   113   by metis
   114 
   115 lemma Quotient_rep_abs:
   116   assumes a: "Quotient R Abs Rep"
   117   shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
   118   using a unfolding Quotient_def
   119   by blast
   120 
   121 lemma Quotient_rel_abs:
   122   assumes a: "Quotient R Abs Rep"
   123   shows "R r s \<Longrightarrow> Abs r = Abs s"
   124   using a unfolding Quotient_def
   125   by blast
   126 
   127 lemma Quotient_symp:
   128   assumes a: "Quotient R Abs Rep"
   129   shows "symp R"
   130   using a unfolding Quotient_def using sympI by metis
   131 
   132 lemma Quotient_transp:
   133   assumes a: "Quotient R Abs Rep"
   134   shows "transp R"
   135   using a unfolding Quotient_def using transpI by metis
   136 
   137 lemma identity_quotient:
   138   shows "Quotient (op =) id id"
   139   unfolding Quotient_def id_def
   140   by blast
   141 
   142 lemma fun_quotient:
   143   assumes q1: "Quotient R1 abs1 rep1"
   144   and     q2: "Quotient R2 abs2 rep2"
   145   shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
   146 proof -
   147   have "\<And>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
   148     using q1 q2 by (simp add: Quotient_def fun_eq_iff)
   149   moreover
   150   have "\<And>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
   151     by (rule fun_relI)
   152       (insert q1 q2 Quotient_rel_abs [of R1 abs1 rep1] Quotient_rel_rep [of R2 abs2 rep2],
   153         simp (no_asm) add: Quotient_def, simp)
   154   moreover
   155   have "\<And>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
   156         (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
   157     apply(auto simp add: fun_rel_def fun_eq_iff)
   158     using q1 q2 unfolding Quotient_def
   159     apply(metis)
   160     using q1 q2 unfolding Quotient_def
   161     apply(metis)
   162     using q1 q2 unfolding Quotient_def
   163     apply(metis)
   164     using q1 q2 unfolding Quotient_def
   165     apply(metis)
   166     done
   167   ultimately
   168   show "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
   169     unfolding Quotient_def by blast
   170 qed
   171 
   172 lemma abs_o_rep:
   173   assumes a: "Quotient R Abs Rep"
   174   shows "Abs o Rep = id"
   175   unfolding fun_eq_iff
   176   by (simp add: Quotient_abs_rep[OF a])
   177 
   178 lemma equals_rsp:
   179   assumes q: "Quotient R Abs Rep"
   180   and     a: "R xa xb" "R ya yb"
   181   shows "R xa ya = R xb yb"
   182   using a Quotient_symp[OF q] Quotient_transp[OF q]
   183   by (blast elim: sympE transpE)
   184 
   185 lemma lambda_prs:
   186   assumes q1: "Quotient R1 Abs1 Rep1"
   187   and     q2: "Quotient R2 Abs2 Rep2"
   188   shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
   189   unfolding fun_eq_iff
   190   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
   191   by simp
   192 
   193 lemma lambda_prs1:
   194   assumes q1: "Quotient R1 Abs1 Rep1"
   195   and     q2: "Quotient R2 Abs2 Rep2"
   196   shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
   197   unfolding fun_eq_iff
   198   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
   199   by simp
   200 
   201 lemma rep_abs_rsp:
   202   assumes q: "Quotient R Abs Rep"
   203   and     a: "R x1 x2"
   204   shows "R x1 (Rep (Abs x2))"
   205   using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
   206   by metis
   207 
   208 lemma rep_abs_rsp_left:
   209   assumes q: "Quotient R Abs Rep"
   210   and     a: "R x1 x2"
   211   shows "R (Rep (Abs x1)) x2"
   212   using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
   213   by metis
   214 
   215 text{*
   216   In the following theorem R1 can be instantiated with anything,
   217   but we know some of the types of the Rep and Abs functions;
   218   so by solving Quotient assumptions we can get a unique R1 that
   219   will be provable; which is why we need to use @{text apply_rsp} and
   220   not the primed version *}
   221 
   222 lemma apply_rsp:
   223   fixes f g::"'a \<Rightarrow> 'c"
   224   assumes q: "Quotient R1 Abs1 Rep1"
   225   and     a: "(R1 ===> R2) f g" "R1 x y"
   226   shows "R2 (f x) (g y)"
   227   using a by (auto elim: fun_relE)
   228 
   229 lemma apply_rsp':
   230   assumes a: "(R1 ===> R2) f g" "R1 x y"
   231   shows "R2 (f x) (g y)"
   232   using a by (auto elim: fun_relE)
   233 
   234 subsection {* lemmas for regularisation of ball and bex *}
   235 
   236 lemma ball_reg_eqv:
   237   fixes P :: "'a \<Rightarrow> bool"
   238   assumes a: "equivp R"
   239   shows "Ball (Respects R) P = (All P)"
   240   using a
   241   unfolding equivp_def
   242   by (auto simp add: in_respects)
   243 
   244 lemma bex_reg_eqv:
   245   fixes P :: "'a \<Rightarrow> bool"
   246   assumes a: "equivp R"
   247   shows "Bex (Respects R) P = (Ex P)"
   248   using a
   249   unfolding equivp_def
   250   by (auto simp add: in_respects)
   251 
   252 lemma ball_reg_right:
   253   assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x"
   254   shows "All P \<longrightarrow> Ball R Q"
   255   using a by (metis Collect_def Collect_mem_eq)
   256 
   257 lemma bex_reg_left:
   258   assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x"
   259   shows "Bex R Q \<longrightarrow> Ex P"
   260   using a by (metis Collect_def Collect_mem_eq)
   261 
   262 lemma ball_reg_left:
   263   assumes a: "equivp R"
   264   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
   265   using a by (metis equivp_reflp in_respects)
   266 
   267 lemma bex_reg_right:
   268   assumes a: "equivp R"
   269   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
   270   using a by (metis equivp_reflp in_respects)
   271 
   272 lemma ball_reg_eqv_range:
   273   fixes P::"'a \<Rightarrow> bool"
   274   and x::"'a"
   275   assumes a: "equivp R2"
   276   shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
   277   apply(rule iffI)
   278   apply(rule allI)
   279   apply(drule_tac x="\<lambda>y. f x" in bspec)
   280   apply(simp add: in_respects fun_rel_def)
   281   apply(rule impI)
   282   using a equivp_reflp_symp_transp[of "R2"]
   283   apply (auto elim: equivpE reflpE)
   284   done
   285 
   286 lemma bex_reg_eqv_range:
   287   assumes a: "equivp R2"
   288   shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
   289   apply(auto)
   290   apply(rule_tac x="\<lambda>y. f x" in bexI)
   291   apply(simp)
   292   apply(simp add: Respects_def in_respects fun_rel_def)
   293   apply(rule impI)
   294   using a equivp_reflp_symp_transp[of "R2"]
   295   apply (auto elim: equivpE reflpE)
   296   done
   297 
   298 (* Next four lemmas are unused *)
   299 lemma all_reg:
   300   assumes a: "!x :: 'a. (P x --> Q x)"
   301   and     b: "All P"
   302   shows "All Q"
   303   using a b by (metis)
   304 
   305 lemma ex_reg:
   306   assumes a: "!x :: 'a. (P x --> Q x)"
   307   and     b: "Ex P"
   308   shows "Ex Q"
   309   using a b by metis
   310 
   311 lemma ball_reg:
   312   assumes a: "!x :: 'a. (R x --> P x --> Q x)"
   313   and     b: "Ball R P"
   314   shows "Ball R Q"
   315   using a b by (metis Collect_def Collect_mem_eq)
   316 
   317 lemma bex_reg:
   318   assumes a: "!x :: 'a. (R x --> P x --> Q x)"
   319   and     b: "Bex R P"
   320   shows "Bex R Q"
   321   using a b by (metis Collect_def Collect_mem_eq)
   322 
   323 
   324 lemma ball_all_comm:
   325   assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
   326   shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
   327   using assms by auto
   328 
   329 lemma bex_ex_comm:
   330   assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
   331   shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
   332   using assms by auto
   333 
   334 subsection {* Bounded abstraction *}
   335 
   336 definition
   337   Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
   338 where
   339   "x \<in> p \<Longrightarrow> Babs p m x = m x"
   340 
   341 lemma babs_rsp:
   342   assumes q: "Quotient R1 Abs1 Rep1"
   343   and     a: "(R1 ===> R2) f g"
   344   shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
   345   apply (auto simp add: Babs_def in_respects fun_rel_def)
   346   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
   347   using a apply (simp add: Babs_def fun_rel_def)
   348   apply (simp add: in_respects fun_rel_def)
   349   using Quotient_rel[OF q]
   350   by metis
   351 
   352 lemma babs_prs:
   353   assumes q1: "Quotient R1 Abs1 Rep1"
   354   and     q2: "Quotient R2 Abs2 Rep2"
   355   shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
   356   apply (rule ext)
   357   apply (simp add:)
   358   apply (subgoal_tac "Rep1 x \<in> Respects R1")
   359   apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
   360   apply (simp add: in_respects Quotient_rel_rep[OF q1])
   361   done
   362 
   363 lemma babs_simp:
   364   assumes q: "Quotient R1 Abs Rep"
   365   shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
   366   apply(rule iffI)
   367   apply(simp_all only: babs_rsp[OF q])
   368   apply(auto simp add: Babs_def fun_rel_def)
   369   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
   370   apply(metis Babs_def)
   371   apply (simp add: in_respects)
   372   using Quotient_rel[OF q]
   373   by metis
   374 
   375 (* If a user proves that a particular functional relation
   376    is an equivalence this may be useful in regularising *)
   377 lemma babs_reg_eqv:
   378   shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
   379   by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
   380 
   381 
   382 (* 3 lemmas needed for proving repabs_inj *)
   383 lemma ball_rsp:
   384   assumes a: "(R ===> (op =)) f g"
   385   shows "Ball (Respects R) f = Ball (Respects R) g"
   386   using a by (auto simp add: Ball_def in_respects elim: fun_relE)
   387 
   388 lemma bex_rsp:
   389   assumes a: "(R ===> (op =)) f g"
   390   shows "(Bex (Respects R) f = Bex (Respects R) g)"
   391   using a by (auto simp add: Bex_def in_respects elim: fun_relE)
   392 
   393 lemma bex1_rsp:
   394   assumes a: "(R ===> (op =)) f g"
   395   shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
   396   using a by (auto elim: fun_relE simp add: Ex1_def in_respects) 
   397 
   398 (* 2 lemmas needed for cleaning of quantifiers *)
   399 lemma all_prs:
   400   assumes a: "Quotient R absf repf"
   401   shows "Ball (Respects R) ((absf ---> id) f) = All f"
   402   using a unfolding Quotient_def Ball_def in_respects id_apply comp_def map_fun_def
   403   by metis
   404 
   405 lemma ex_prs:
   406   assumes a: "Quotient R absf repf"
   407   shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
   408   using a unfolding Quotient_def Bex_def in_respects id_apply comp_def map_fun_def
   409   by metis
   410 
   411 subsection {* @{text Bex1_rel} quantifier *}
   412 
   413 definition
   414   Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
   415 where
   416   "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)))"
   417 
   418 lemma bex1_rel_aux:
   419   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
   420   unfolding Bex1_rel_def
   421   apply (erule conjE)+
   422   apply (erule bexE)
   423   apply rule
   424   apply (rule_tac x="xa" in bexI)
   425   apply metis
   426   apply metis
   427   apply rule+
   428   apply (erule_tac x="xaa" in ballE)
   429   prefer 2
   430   apply (metis)
   431   apply (erule_tac x="ya" in ballE)
   432   prefer 2
   433   apply (metis)
   434   apply (metis in_respects)
   435   done
   436 
   437 lemma bex1_rel_aux2:
   438   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
   439   unfolding Bex1_rel_def
   440   apply (erule conjE)+
   441   apply (erule bexE)
   442   apply rule
   443   apply (rule_tac x="xa" in bexI)
   444   apply metis
   445   apply metis
   446   apply rule+
   447   apply (erule_tac x="xaa" in ballE)
   448   prefer 2
   449   apply (metis)
   450   apply (erule_tac x="ya" in ballE)
   451   prefer 2
   452   apply (metis)
   453   apply (metis in_respects)
   454   done
   455 
   456 lemma bex1_rel_rsp:
   457   assumes a: "Quotient R absf repf"
   458   shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
   459   apply (simp add: fun_rel_def)
   460   apply clarify
   461   apply rule
   462   apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
   463   apply (erule bex1_rel_aux2)
   464   apply assumption
   465   done
   466 
   467 
   468 lemma ex1_prs:
   469   assumes a: "Quotient R absf repf"
   470   shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
   471 apply (simp add:)
   472 apply (subst Bex1_rel_def)
   473 apply (subst Bex_def)
   474 apply (subst Ex1_def)
   475 apply simp
   476 apply rule
   477  apply (erule conjE)+
   478  apply (erule_tac exE)
   479  apply (erule conjE)
   480  apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
   481   apply (rule_tac x="absf x" in exI)
   482   apply (simp)
   483   apply rule+
   484   using a unfolding Quotient_def
   485   apply metis
   486  apply rule+
   487  apply (erule_tac x="x" in ballE)
   488   apply (erule_tac x="y" in ballE)
   489    apply simp
   490   apply (simp add: in_respects)
   491  apply (simp add: in_respects)
   492 apply (erule_tac exE)
   493  apply rule
   494  apply (rule_tac x="repf x" in exI)
   495  apply (simp only: in_respects)
   496   apply rule
   497  apply (metis Quotient_rel_rep[OF a])
   498 using a unfolding Quotient_def apply (simp)
   499 apply rule+
   500 using a unfolding Quotient_def in_respects
   501 apply metis
   502 done
   503 
   504 lemma bex1_bexeq_reg:
   505   shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
   506   apply (simp add: Ex1_def Bex1_rel_def in_respects)
   507   apply clarify
   508   apply auto
   509   apply (rule bexI)
   510   apply assumption
   511   apply (simp add: in_respects)
   512   apply (simp add: in_respects)
   513   apply auto
   514   done
   515 
   516 lemma bex1_bexeq_reg_eqv:
   517   assumes a: "equivp R"
   518   shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
   519   using equivp_reflp[OF a]
   520   apply (intro impI)
   521   apply (elim ex1E)
   522   apply (rule mp[OF bex1_bexeq_reg])
   523   apply (rule_tac a="x" in ex1I)
   524   apply (subst in_respects)
   525   apply (rule conjI)
   526   apply assumption
   527   apply assumption
   528   apply clarify
   529   apply (erule_tac x="xa" in allE)
   530   apply simp
   531   done
   532 
   533 subsection {* Various respects and preserve lemmas *}
   534 
   535 lemma quot_rel_rsp:
   536   assumes a: "Quotient R Abs Rep"
   537   shows "(R ===> R ===> op =) R R"
   538   apply(rule fun_relI)+
   539   apply(rule equals_rsp[OF a])
   540   apply(assumption)+
   541   done
   542 
   543 lemma o_prs:
   544   assumes q1: "Quotient R1 Abs1 Rep1"
   545   and     q2: "Quotient R2 Abs2 Rep2"
   546   and     q3: "Quotient R3 Abs3 Rep3"
   547   shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
   548   and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
   549   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3]
   550   by (simp_all add: fun_eq_iff)
   551 
   552 lemma o_rsp:
   553   "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
   554   "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
   555   by (auto intro!: fun_relI elim: fun_relE)
   556 
   557 lemma cond_prs:
   558   assumes a: "Quotient R absf repf"
   559   shows "absf (if a then repf b else repf c) = (if a then b else c)"
   560   using a unfolding Quotient_def by auto
   561 
   562 lemma if_prs:
   563   assumes q: "Quotient R Abs Rep"
   564   shows "(id ---> Rep ---> Rep ---> Abs) If = If"
   565   using Quotient_abs_rep[OF q]
   566   by (auto simp add: fun_eq_iff)
   567 
   568 lemma if_rsp:
   569   assumes q: "Quotient R Abs Rep"
   570   shows "(op = ===> R ===> R ===> R) If If"
   571   by (auto intro!: fun_relI)
   572 
   573 lemma let_prs:
   574   assumes q1: "Quotient R1 Abs1 Rep1"
   575   and     q2: "Quotient R2 Abs2 Rep2"
   576   shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
   577   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
   578   by (auto simp add: fun_eq_iff)
   579 
   580 lemma let_rsp:
   581   shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
   582   by (auto intro!: fun_relI elim: fun_relE)
   583 
   584 lemma mem_rsp:
   585   shows "(R1 ===> (R1 ===> R2) ===> R2) op \<in> op \<in>"
   586   by (auto intro!: fun_relI elim: fun_relE simp add: mem_def)
   587 
   588 lemma mem_prs:
   589   assumes a1: "Quotient R1 Abs1 Rep1"
   590   and     a2: "Quotient R2 Abs2 Rep2"
   591   shows "(Rep1 ---> (Abs1 ---> Rep2) ---> Abs2) op \<in> = op \<in>"
   592   by (simp add: fun_eq_iff mem_def Quotient_abs_rep[OF a1] Quotient_abs_rep[OF a2])
   593 
   594 lemma id_rsp:
   595   shows "(R ===> R) id id"
   596   by (auto intro: fun_relI)
   597 
   598 lemma id_prs:
   599   assumes a: "Quotient R Abs Rep"
   600   shows "(Rep ---> Abs) id = id"
   601   by (simp add: fun_eq_iff Quotient_abs_rep [OF a])
   602 
   603 
   604 locale quot_type =
   605   fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
   606   and   Abs :: "('a \<Rightarrow> bool) \<Rightarrow> 'b"
   607   and   Rep :: "'b \<Rightarrow> ('a \<Rightarrow> bool)"
   608   assumes equivp: "part_equivp R"
   609   and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = R x"
   610   and     rep_inverse: "\<And>x. Abs (Rep x) = x"
   611   and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = R x))) \<Longrightarrow> (Rep (Abs c)) = c"
   612   and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
   613 begin
   614 
   615 definition
   616   abs :: "'a \<Rightarrow> 'b"
   617 where
   618   "abs x = Abs (R x)"
   619 
   620 definition
   621   rep :: "'b \<Rightarrow> 'a"
   622 where
   623   "rep a = Eps (Rep a)"
   624 
   625 lemma homeier5:
   626   assumes a: "R r r"
   627   shows "Rep (Abs (R r)) = R r"
   628   apply (subst abs_inverse)
   629   using a by auto
   630 
   631 theorem homeier6:
   632   assumes a: "R r r"
   633   and b: "R s s"
   634   shows "Abs (R r) = Abs (R s) \<longleftrightarrow> R r = R s"
   635   by (metis a b homeier5)
   636 
   637 theorem homeier8:
   638   assumes "R r r"
   639   shows "R (Eps (R r)) = R r"
   640   using assms equivp[simplified part_equivp_def]
   641   apply clarify
   642   by (metis assms exE_some)
   643 
   644 lemma Quotient:
   645   shows "Quotient R abs rep"
   646   unfolding Quotient_def abs_def rep_def
   647   proof (intro conjI allI)
   648     fix a r s
   649     show "Abs (R (Eps (Rep a))) = a"
   650       by (metis equivp exE_some part_equivp_def rep_inverse rep_prop)
   651     show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (R r) = Abs (R s))"
   652       by (metis homeier6 equivp[simplified part_equivp_def])
   653     show "R (Eps (Rep a)) (Eps (Rep a))" proof -
   654       obtain x where r: "R x x" and rep: "Rep a = R x" using rep_prop[of a] by auto
   655       have "R (Eps (R x)) x" using homeier8 r by simp
   656       then have "R x (Eps (R x))" using part_equivp_symp[OF equivp] by fast
   657       then have "R (Eps (R x)) (Eps (R x))" using homeier8[OF r] by simp
   658       then show "R (Eps (Rep a)) (Eps (Rep a))" using rep by simp
   659     qed
   660   qed
   661 
   662 end
   663 
   664 
   665 subsection {* ML setup *}
   666 
   667 text {* Auxiliary data for the quotient package *}
   668 
   669 use "Tools/Quotient/quotient_info.ML"
   670 setup Quotient_Info.setup
   671 
   672 declare [[map "fun" = (map_fun, fun_rel)]]
   673 
   674 lemmas [quot_thm] = fun_quotient
   675 lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp mem_rsp id_rsp
   676 lemmas [quot_preserve] = if_prs o_prs let_prs mem_prs id_prs
   677 lemmas [quot_equiv] = identity_equivp
   678 
   679 
   680 text {* Lemmas about simplifying id's. *}
   681 lemmas [id_simps] =
   682   id_def[symmetric]
   683   map_fun_id
   684   id_apply
   685   id_o
   686   o_id
   687   eq_comp_r
   688 
   689 text {* Translation functions for the lifting process. *}
   690 use "Tools/Quotient/quotient_term.ML"
   691 
   692 
   693 text {* Definitions of the quotient types. *}
   694 use "Tools/Quotient/quotient_typ.ML"
   695 
   696 
   697 text {* Definitions for quotient constants. *}
   698 use "Tools/Quotient/quotient_def.ML"
   699 
   700 
   701 text {*
   702   An auxiliary constant for recording some information
   703   about the lifted theorem in a tactic.
   704 *}
   705 definition
   706   Quot_True :: "'a \<Rightarrow> bool"
   707 where
   708   "Quot_True x \<longleftrightarrow> True"
   709 
   710 lemma
   711   shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
   712   and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
   713   and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
   714   and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
   715   and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
   716   by (simp_all add: Quot_True_def ext)
   717 
   718 lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
   719   by (simp add: Quot_True_def)
   720 
   721 
   722 text {* Tactics for proving the lifted theorems *}
   723 use "Tools/Quotient/quotient_tacs.ML"
   724 
   725 subsection {* Methods / Interface *}
   726 
   727 method_setup lifting =
   728   {* Attrib.thms >> (fn thms => fn ctxt => 
   729        SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_tac ctxt [] thms))) *}
   730   {* lifts theorems to quotient types *}
   731 
   732 method_setup lifting_setup =
   733   {* Attrib.thm >> (fn thm => fn ctxt => 
   734        SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_procedure_tac ctxt [] thm))) *}
   735   {* sets up the three goals for the quotient lifting procedure *}
   736 
   737 method_setup descending =
   738   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.descend_tac ctxt []))) *}
   739   {* decends theorems to the raw level *}
   740 
   741 method_setup descending_setup =
   742   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.descend_procedure_tac ctxt []))) *}
   743   {* sets up the three goals for the decending theorems *}
   744 
   745 method_setup regularize =
   746   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.regularize_tac ctxt))) *}
   747   {* proves the regularization goals from the quotient lifting procedure *}
   748 
   749 method_setup injection =
   750   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.all_injection_tac ctxt))) *}
   751   {* proves the rep/abs injection goals from the quotient lifting procedure *}
   752 
   753 method_setup cleaning =
   754   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.clean_tac ctxt))) *}
   755   {* proves the cleaning goals from the quotient lifting procedure *}
   756 
   757 attribute_setup quot_lifted =
   758   {* Scan.succeed Quotient_Tacs.lifted_attrib *}
   759   {* lifts theorems to quotient types *}
   760 
   761 no_notation
   762   rel_conj (infixr "OOO" 75) and
   763   map_fun (infixr "--->" 55) and
   764   fun_rel (infixr "===>" 55)
   765 
   766 end