src/HOL/Quotient.thy
 author haftmann Thu Nov 18 17:01:15 2010 +0100 (2010-11-18) changeset 40602 91e583511113 parent 40466 c6587375088e child 40615 ab551d108feb permissions -rw-r--r--
map_fun combinator in theory Fun
```     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 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
```
```    18 text {*
```
```    19   Basic definition for equivalence relations
```
```    20   that are represented by predicates.
```
```    21 *}
```
```    22
```
```    23 definition
```
```    24   "reflp E \<longleftrightarrow> (\<forall>x. E x x)"
```
```    25
```
```    26 lemma refl_reflp:
```
```    27   "refl A \<longleftrightarrow> reflp (\<lambda>x y. (x, y) \<in> A)"
```
```    28   by (simp add: refl_on_def reflp_def)
```
```    29
```
```    30 definition
```
```    31   "symp E \<longleftrightarrow> (\<forall>x y. E x y \<longrightarrow> E y x)"
```
```    32
```
```    33 lemma sym_symp:
```
```    34   "sym A \<longleftrightarrow> symp (\<lambda>x y. (x, y) \<in> A)"
```
```    35   by (simp add: sym_def symp_def)
```
```    36
```
```    37 definition
```
```    38   "transp E \<longleftrightarrow> (\<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z)"
```
```    39
```
```    40 lemma trans_transp:
```
```    41   "trans A \<longleftrightarrow> transp (\<lambda>x y. (x, y) \<in> A)"
```
```    42   by (auto simp add: trans_def transp_def)
```
```    43
```
```    44 definition
```
```    45   "equivp E \<longleftrightarrow> (\<forall>x y. E x y = (E x = E y))"
```
```    46
```
```    47 lemma equivp_reflp_symp_transp:
```
```    48   shows "equivp E = (reflp E \<and> symp E \<and> transp E)"
```
```    49   unfolding equivp_def reflp_def symp_def transp_def fun_eq_iff
```
```    50   by blast
```
```    51
```
```    52 lemma equiv_equivp:
```
```    53   "equiv UNIV A \<longleftrightarrow> equivp (\<lambda>x y. (x, y) \<in> A)"
```
```    54   by (simp add: equiv_def equivp_reflp_symp_transp refl_reflp sym_symp trans_transp)
```
```    55
```
```    56 lemma equivp_reflp:
```
```    57   shows "equivp E \<Longrightarrow> E x x"
```
```    58   by (simp only: equivp_reflp_symp_transp reflp_def)
```
```    59
```
```    60 lemma equivp_symp:
```
```    61   shows "equivp E \<Longrightarrow> E x y \<Longrightarrow> E y x"
```
```    62   by (simp add: equivp_def)
```
```    63
```
```    64 lemma equivp_transp:
```
```    65   shows "equivp E \<Longrightarrow> E x y \<Longrightarrow> E y z \<Longrightarrow> E x z"
```
```    66   by (simp add: equivp_def)
```
```    67
```
```    68 lemma equivpI:
```
```    69   assumes "reflp R" "symp R" "transp R"
```
```    70   shows "equivp R"
```
```    71   using assms by (simp add: equivp_reflp_symp_transp)
```
```    72
```
```    73 lemma identity_equivp:
```
```    74   shows "equivp (op =)"
```
```    75   unfolding equivp_def
```
```    76   by auto
```
```    77
```
```    78 text {* Partial equivalences *}
```
```    79
```
```    80 definition
```
```    81   "part_equivp E \<longleftrightarrow> (\<exists>x. E x x) \<and> (\<forall>x y. E x y = (E x x \<and> E y y \<and> (E x = E y)))"
```
```    82
```
```    83 lemma equivp_implies_part_equivp:
```
```    84   assumes a: "equivp E"
```
```    85   shows "part_equivp E"
```
```    86   using a
```
```    87   unfolding equivp_def part_equivp_def
```
```    88   by auto
```
```    89
```
```    90 lemma part_equivp_symp:
```
```    91   assumes e: "part_equivp R"
```
```    92   and a: "R x y"
```
```    93   shows "R y x"
```
```    94   using e[simplified part_equivp_def] a
```
```    95   by (metis)
```
```    96
```
```    97 lemma part_equivp_typedef:
```
```    98   shows "part_equivp R \<Longrightarrow> \<exists>d. d \<in> (\<lambda>c. \<exists>x. R x x \<and> c = R x)"
```
```    99   unfolding part_equivp_def mem_def
```
```   100   apply clarify
```
```   101   apply (intro exI)
```
```   102   apply (rule conjI)
```
```   103   apply assumption
```
```   104   apply (rule refl)
```
```   105   done
```
```   106
```
```   107 lemma part_equivp_refl_symp_transp:
```
```   108   shows "part_equivp E \<longleftrightarrow> ((\<exists>x. E x x) \<and> symp E \<and> transp E)"
```
```   109 proof
```
```   110   assume "part_equivp E"
```
```   111   then show "(\<exists>x. E x x) \<and> symp E \<and> transp E"
```
```   112   unfolding part_equivp_def symp_def transp_def
```
```   113   by metis
```
```   114 next
```
```   115   assume a: "(\<exists>x. E x x) \<and> symp E \<and> transp E"
```
```   116   then have b: "(\<forall>x y. E x y \<longrightarrow> E y x)" and c: "(\<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z)"
```
```   117     unfolding symp_def transp_def by (metis, metis)
```
```   118   have "(\<forall>x y. E x y = (E x x \<and> E y y \<and> E x = E y))"
```
```   119   proof (intro allI iffI conjI)
```
```   120     fix x y
```
```   121     assume d: "E x y"
```
```   122     then show "E x x" using b c by metis
```
```   123     show "E y y" using b c d by metis
```
```   124     show "E x = E y" unfolding fun_eq_iff using b c d by metis
```
```   125   next
```
```   126     fix x y
```
```   127     assume "E x x \<and> E y y \<and> E x = E y"
```
```   128     then show "E x y" using b c by metis
```
```   129   qed
```
```   130   then show "part_equivp E" unfolding part_equivp_def using a by metis
```
```   131 qed
```
```   132
```
```   133 lemma part_equivpI:
```
```   134   assumes "\<exists>x. R x x" "symp R" "transp R"
```
```   135   shows "part_equivp R"
```
```   136   using assms by (simp add: part_equivp_refl_symp_transp)
```
```   137
```
```   138 text {* Composition of Relations *}
```
```   139
```
```   140 abbreviation
```
```   141   rel_conj (infixr "OOO" 75)
```
```   142 where
```
```   143   "r1 OOO r2 \<equiv> r1 OO r2 OO r1"
```
```   144
```
```   145 lemma eq_comp_r:
```
```   146   shows "((op =) OOO R) = R"
```
```   147   by (auto simp add: fun_eq_iff)
```
```   148
```
```   149 subsection {* Respects predicate *}
```
```   150
```
```   151 definition
```
```   152   Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
```
```   153 where
```
```   154   "Respects R x = R x x"
```
```   155
```
```   156 lemma in_respects:
```
```   157   shows "x \<in> Respects R \<longleftrightarrow> R x x"
```
```   158   unfolding mem_def Respects_def
```
```   159   by simp
```
```   160
```
```   161 subsection {* Function map and function relation *}
```
```   162
```
```   163 notation map_fun (infixr "--->" 55)
```
```   164
```
```   165 lemma map_fun_id:
```
```   166   "(id ---> id) = id"
```
```   167   by (simp add: fun_eq_iff)
```
```   168
```
```   169 definition
```
```   170   fun_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" (infixr "===>" 55)
```
```   171 where
```
```   172   "fun_rel E1 E2 = (\<lambda>f g. \<forall>x y. E1 x y \<longrightarrow> E2 (f x) (g y))"
```
```   173
```
```   174 lemma fun_relI [intro]:
```
```   175   assumes "\<And>x y. E1 x y \<Longrightarrow> E2 (f x) (g y)"
```
```   176   shows "(E1 ===> E2) f g"
```
```   177   using assms by (simp add: fun_rel_def)
```
```   178
```
```   179 lemma fun_relE:
```
```   180   assumes "(E1 ===> E2) f g" and "E1 x y"
```
```   181   obtains "E2 (f x) (g y)"
```
```   182   using assms by (simp add: fun_rel_def)
```
```   183
```
```   184 lemma fun_rel_eq:
```
```   185   shows "((op =) ===> (op =)) = (op =)"
```
```   186   by (auto simp add: fun_eq_iff elim: fun_relE)
```
```   187
```
```   188
```
```   189 subsection {* Quotient Predicate *}
```
```   190
```
```   191 definition
```
```   192   "Quotient E Abs Rep \<longleftrightarrow>
```
```   193      (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. E (Rep a) (Rep a)) \<and>
```
```   194      (\<forall>r s. E r s = (E r r \<and> E s s \<and> (Abs r = Abs s)))"
```
```   195
```
```   196 lemma Quotient_abs_rep:
```
```   197   assumes a: "Quotient E Abs Rep"
```
```   198   shows "Abs (Rep a) = a"
```
```   199   using a
```
```   200   unfolding Quotient_def
```
```   201   by simp
```
```   202
```
```   203 lemma Quotient_rep_reflp:
```
```   204   assumes a: "Quotient E Abs Rep"
```
```   205   shows "E (Rep a) (Rep a)"
```
```   206   using a
```
```   207   unfolding Quotient_def
```
```   208   by blast
```
```   209
```
```   210 lemma Quotient_rel:
```
```   211   assumes a: "Quotient E Abs Rep"
```
```   212   shows " E r s = (E r r \<and> E s s \<and> (Abs r = Abs s))"
```
```   213   using a
```
```   214   unfolding Quotient_def
```
```   215   by blast
```
```   216
```
```   217 lemma Quotient_rel_rep:
```
```   218   assumes a: "Quotient R Abs Rep"
```
```   219   shows "R (Rep a) (Rep b) = (a = b)"
```
```   220   using a
```
```   221   unfolding Quotient_def
```
```   222   by metis
```
```   223
```
```   224 lemma Quotient_rep_abs:
```
```   225   assumes a: "Quotient R Abs Rep"
```
```   226   shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
```
```   227   using a unfolding Quotient_def
```
```   228   by blast
```
```   229
```
```   230 lemma Quotient_rel_abs:
```
```   231   assumes a: "Quotient E Abs Rep"
```
```   232   shows "E r s \<Longrightarrow> Abs r = Abs s"
```
```   233   using a unfolding Quotient_def
```
```   234   by blast
```
```   235
```
```   236 lemma Quotient_symp:
```
```   237   assumes a: "Quotient E Abs Rep"
```
```   238   shows "symp E"
```
```   239   using a unfolding Quotient_def symp_def
```
```   240   by metis
```
```   241
```
```   242 lemma Quotient_transp:
```
```   243   assumes a: "Quotient E Abs Rep"
```
```   244   shows "transp E"
```
```   245   using a unfolding Quotient_def transp_def
```
```   246   by metis
```
```   247
```
```   248 lemma identity_quotient:
```
```   249   shows "Quotient (op =) id id"
```
```   250   unfolding Quotient_def id_def
```
```   251   by blast
```
```   252
```
```   253 lemma fun_quotient:
```
```   254   assumes q1: "Quotient R1 abs1 rep1"
```
```   255   and     q2: "Quotient R2 abs2 rep2"
```
```   256   shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
```
```   257 proof -
```
```   258   have "\<And>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
```
```   259     using q1 q2 by (simp add: Quotient_def fun_eq_iff)
```
```   260   moreover
```
```   261   have "\<And>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
```
```   262     by (rule fun_relI)
```
```   263       (insert q1 q2 Quotient_rel_abs [of R1 abs1 rep1] Quotient_rel_rep [of R2 abs2 rep2],
```
```   264         simp (no_asm) add: Quotient_def, simp)
```
```   265   moreover
```
```   266   have "\<And>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
```
```   267         (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
```
```   268     apply(auto simp add: fun_rel_def fun_eq_iff)
```
```   269     using q1 q2 unfolding Quotient_def
```
```   270     apply(metis)
```
```   271     using q1 q2 unfolding Quotient_def
```
```   272     apply(metis)
```
```   273     using q1 q2 unfolding Quotient_def
```
```   274     apply(metis)
```
```   275     using q1 q2 unfolding Quotient_def
```
```   276     apply(metis)
```
```   277     done
```
```   278   ultimately
```
```   279   show "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
```
```   280     unfolding Quotient_def by blast
```
```   281 qed
```
```   282
```
```   283 lemma abs_o_rep:
```
```   284   assumes a: "Quotient R Abs Rep"
```
```   285   shows "Abs o Rep = id"
```
```   286   unfolding fun_eq_iff
```
```   287   by (simp add: Quotient_abs_rep[OF a])
```
```   288
```
```   289 lemma equals_rsp:
```
```   290   assumes q: "Quotient R Abs Rep"
```
```   291   and     a: "R xa xb" "R ya yb"
```
```   292   shows "R xa ya = R xb yb"
```
```   293   using a Quotient_symp[OF q] Quotient_transp[OF q]
```
```   294   unfolding symp_def transp_def
```
```   295   by blast
```
```   296
```
```   297 lemma lambda_prs:
```
```   298   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   299   and     q2: "Quotient R2 Abs2 Rep2"
```
```   300   shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
```
```   301   unfolding fun_eq_iff
```
```   302   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
```
```   303   by (simp add:)
```
```   304
```
```   305 lemma lambda_prs1:
```
```   306   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   307   and     q2: "Quotient R2 Abs2 Rep2"
```
```   308   shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
```
```   309   unfolding fun_eq_iff
```
```   310   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
```
```   311   by (simp add:)
```
```   312
```
```   313 lemma rep_abs_rsp:
```
```   314   assumes q: "Quotient R Abs Rep"
```
```   315   and     a: "R x1 x2"
```
```   316   shows "R x1 (Rep (Abs x2))"
```
```   317   using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
```
```   318   by metis
```
```   319
```
```   320 lemma rep_abs_rsp_left:
```
```   321   assumes q: "Quotient R Abs Rep"
```
```   322   and     a: "R x1 x2"
```
```   323   shows "R (Rep (Abs x1)) x2"
```
```   324   using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
```
```   325   by metis
```
```   326
```
```   327 text{*
```
```   328   In the following theorem R1 can be instantiated with anything,
```
```   329   but we know some of the types of the Rep and Abs functions;
```
```   330   so by solving Quotient assumptions we can get a unique R1 that
```
```   331   will be provable; which is why we need to use @{text apply_rsp} and
```
```   332   not the primed version *}
```
```   333
```
```   334 lemma apply_rsp:
```
```   335   fixes f g::"'a \<Rightarrow> 'c"
```
```   336   assumes q: "Quotient R1 Abs1 Rep1"
```
```   337   and     a: "(R1 ===> R2) f g" "R1 x y"
```
```   338   shows "R2 (f x) (g y)"
```
```   339   using a by (auto elim: fun_relE)
```
```   340
```
```   341 lemma apply_rsp':
```
```   342   assumes a: "(R1 ===> R2) f g" "R1 x y"
```
```   343   shows "R2 (f x) (g y)"
```
```   344   using a by (auto elim: fun_relE)
```
```   345
```
```   346 subsection {* lemmas for regularisation of ball and bex *}
```
```   347
```
```   348 lemma ball_reg_eqv:
```
```   349   fixes P :: "'a \<Rightarrow> bool"
```
```   350   assumes a: "equivp R"
```
```   351   shows "Ball (Respects R) P = (All P)"
```
```   352   using a
```
```   353   unfolding equivp_def
```
```   354   by (auto simp add: in_respects)
```
```   355
```
```   356 lemma bex_reg_eqv:
```
```   357   fixes P :: "'a \<Rightarrow> bool"
```
```   358   assumes a: "equivp R"
```
```   359   shows "Bex (Respects R) P = (Ex P)"
```
```   360   using a
```
```   361   unfolding equivp_def
```
```   362   by (auto simp add: in_respects)
```
```   363
```
```   364 lemma ball_reg_right:
```
```   365   assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x"
```
```   366   shows "All P \<longrightarrow> Ball R Q"
```
```   367   using a by (metis Collect_def Collect_mem_eq)
```
```   368
```
```   369 lemma bex_reg_left:
```
```   370   assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x"
```
```   371   shows "Bex R Q \<longrightarrow> Ex P"
```
```   372   using a by (metis Collect_def Collect_mem_eq)
```
```   373
```
```   374 lemma ball_reg_left:
```
```   375   assumes a: "equivp R"
```
```   376   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
```
```   377   using a by (metis equivp_reflp in_respects)
```
```   378
```
```   379 lemma bex_reg_right:
```
```   380   assumes a: "equivp R"
```
```   381   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
```
```   382   using a by (metis equivp_reflp in_respects)
```
```   383
```
```   384 lemma ball_reg_eqv_range:
```
```   385   fixes P::"'a \<Rightarrow> bool"
```
```   386   and x::"'a"
```
```   387   assumes a: "equivp R2"
```
```   388   shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
```
```   389   apply(rule iffI)
```
```   390   apply(rule allI)
```
```   391   apply(drule_tac x="\<lambda>y. f x" in bspec)
```
```   392   apply(simp add: in_respects fun_rel_def)
```
```   393   apply(rule impI)
```
```   394   using a equivp_reflp_symp_transp[of "R2"]
```
```   395   apply(simp add: reflp_def)
```
```   396   apply(simp)
```
```   397   apply(simp)
```
```   398   done
```
```   399
```
```   400 lemma bex_reg_eqv_range:
```
```   401   assumes a: "equivp R2"
```
```   402   shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
```
```   403   apply(auto)
```
```   404   apply(rule_tac x="\<lambda>y. f x" in bexI)
```
```   405   apply(simp)
```
```   406   apply(simp add: Respects_def in_respects fun_rel_def)
```
```   407   apply(rule impI)
```
```   408   using a equivp_reflp_symp_transp[of "R2"]
```
```   409   apply(simp add: reflp_def)
```
```   410   done
```
```   411
```
```   412 (* Next four lemmas are unused *)
```
```   413 lemma all_reg:
```
```   414   assumes a: "!x :: 'a. (P x --> Q x)"
```
```   415   and     b: "All P"
```
```   416   shows "All Q"
```
```   417   using a b by (metis)
```
```   418
```
```   419 lemma ex_reg:
```
```   420   assumes a: "!x :: 'a. (P x --> Q x)"
```
```   421   and     b: "Ex P"
```
```   422   shows "Ex Q"
```
```   423   using a b by metis
```
```   424
```
```   425 lemma ball_reg:
```
```   426   assumes a: "!x :: 'a. (R x --> P x --> Q x)"
```
```   427   and     b: "Ball R P"
```
```   428   shows "Ball R Q"
```
```   429   using a b by (metis Collect_def Collect_mem_eq)
```
```   430
```
```   431 lemma bex_reg:
```
```   432   assumes a: "!x :: 'a. (R x --> P x --> Q x)"
```
```   433   and     b: "Bex R P"
```
```   434   shows "Bex R Q"
```
```   435   using a b by (metis Collect_def Collect_mem_eq)
```
```   436
```
```   437
```
```   438 lemma ball_all_comm:
```
```   439   assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
```
```   440   shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
```
```   441   using assms by auto
```
```   442
```
```   443 lemma bex_ex_comm:
```
```   444   assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
```
```   445   shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
```
```   446   using assms by auto
```
```   447
```
```   448 subsection {* Bounded abstraction *}
```
```   449
```
```   450 definition
```
```   451   Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
```
```   452 where
```
```   453   "x \<in> p \<Longrightarrow> Babs p m x = m x"
```
```   454
```
```   455 lemma babs_rsp:
```
```   456   assumes q: "Quotient R1 Abs1 Rep1"
```
```   457   and     a: "(R1 ===> R2) f g"
```
```   458   shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
```
```   459   apply (auto simp add: Babs_def in_respects fun_rel_def)
```
```   460   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
```
```   461   using a apply (simp add: Babs_def fun_rel_def)
```
```   462   apply (simp add: in_respects fun_rel_def)
```
```   463   using Quotient_rel[OF q]
```
```   464   by metis
```
```   465
```
```   466 lemma babs_prs:
```
```   467   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   468   and     q2: "Quotient R2 Abs2 Rep2"
```
```   469   shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
```
```   470   apply (rule ext)
```
```   471   apply (simp add:)
```
```   472   apply (subgoal_tac "Rep1 x \<in> Respects R1")
```
```   473   apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
```
```   474   apply (simp add: in_respects Quotient_rel_rep[OF q1])
```
```   475   done
```
```   476
```
```   477 lemma babs_simp:
```
```   478   assumes q: "Quotient R1 Abs Rep"
```
```   479   shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
```
```   480   apply(rule iffI)
```
```   481   apply(simp_all only: babs_rsp[OF q])
```
```   482   apply(auto simp add: Babs_def fun_rel_def)
```
```   483   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
```
```   484   apply(metis Babs_def)
```
```   485   apply (simp add: in_respects)
```
```   486   using Quotient_rel[OF q]
```
```   487   by metis
```
```   488
```
```   489 (* If a user proves that a particular functional relation
```
```   490    is an equivalence this may be useful in regularising *)
```
```   491 lemma babs_reg_eqv:
```
```   492   shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
```
```   493   by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
```
```   494
```
```   495
```
```   496 (* 3 lemmas needed for proving repabs_inj *)
```
```   497 lemma ball_rsp:
```
```   498   assumes a: "(R ===> (op =)) f g"
```
```   499   shows "Ball (Respects R) f = Ball (Respects R) g"
```
```   500   using a by (auto simp add: Ball_def in_respects elim: fun_relE)
```
```   501
```
```   502 lemma bex_rsp:
```
```   503   assumes a: "(R ===> (op =)) f g"
```
```   504   shows "(Bex (Respects R) f = Bex (Respects R) g)"
```
```   505   using a by (auto simp add: Bex_def in_respects elim: fun_relE)
```
```   506
```
```   507 lemma bex1_rsp:
```
```   508   assumes a: "(R ===> (op =)) f g"
```
```   509   shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
```
```   510   using a by (auto elim: fun_relE simp add: Ex1_def in_respects)
```
```   511
```
```   512 (* 2 lemmas needed for cleaning of quantifiers *)
```
```   513 lemma all_prs:
```
```   514   assumes a: "Quotient R absf repf"
```
```   515   shows "Ball (Respects R) ((absf ---> id) f) = All f"
```
```   516   using a unfolding Quotient_def Ball_def in_respects id_apply comp_def map_fun_def
```
```   517   by metis
```
```   518
```
```   519 lemma ex_prs:
```
```   520   assumes a: "Quotient R absf repf"
```
```   521   shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
```
```   522   using a unfolding Quotient_def Bex_def in_respects id_apply comp_def map_fun_def
```
```   523   by metis
```
```   524
```
```   525 subsection {* @{text Bex1_rel} quantifier *}
```
```   526
```
```   527 definition
```
```   528   Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```   529 where
```
```   530   "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)))"
```
```   531
```
```   532 lemma bex1_rel_aux:
```
```   533   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
```
```   534   unfolding Bex1_rel_def
```
```   535   apply (erule conjE)+
```
```   536   apply (erule bexE)
```
```   537   apply rule
```
```   538   apply (rule_tac x="xa" in bexI)
```
```   539   apply metis
```
```   540   apply metis
```
```   541   apply rule+
```
```   542   apply (erule_tac x="xaa" in ballE)
```
```   543   prefer 2
```
```   544   apply (metis)
```
```   545   apply (erule_tac x="ya" in ballE)
```
```   546   prefer 2
```
```   547   apply (metis)
```
```   548   apply (metis in_respects)
```
```   549   done
```
```   550
```
```   551 lemma bex1_rel_aux2:
```
```   552   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
```
```   553   unfolding Bex1_rel_def
```
```   554   apply (erule conjE)+
```
```   555   apply (erule bexE)
```
```   556   apply rule
```
```   557   apply (rule_tac x="xa" in bexI)
```
```   558   apply metis
```
```   559   apply metis
```
```   560   apply rule+
```
```   561   apply (erule_tac x="xaa" in ballE)
```
```   562   prefer 2
```
```   563   apply (metis)
```
```   564   apply (erule_tac x="ya" in ballE)
```
```   565   prefer 2
```
```   566   apply (metis)
```
```   567   apply (metis in_respects)
```
```   568   done
```
```   569
```
```   570 lemma bex1_rel_rsp:
```
```   571   assumes a: "Quotient R absf repf"
```
```   572   shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
```
```   573   apply (simp add: fun_rel_def)
```
```   574   apply clarify
```
```   575   apply rule
```
```   576   apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
```
```   577   apply (erule bex1_rel_aux2)
```
```   578   apply assumption
```
```   579   done
```
```   580
```
```   581
```
```   582 lemma ex1_prs:
```
```   583   assumes a: "Quotient R absf repf"
```
```   584   shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
```
```   585 apply (simp add:)
```
```   586 apply (subst Bex1_rel_def)
```
```   587 apply (subst Bex_def)
```
```   588 apply (subst Ex1_def)
```
```   589 apply simp
```
```   590 apply rule
```
```   591  apply (erule conjE)+
```
```   592  apply (erule_tac exE)
```
```   593  apply (erule conjE)
```
```   594  apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
```
```   595   apply (rule_tac x="absf x" in exI)
```
```   596   apply (simp)
```
```   597   apply rule+
```
```   598   using a unfolding Quotient_def
```
```   599   apply metis
```
```   600  apply rule+
```
```   601  apply (erule_tac x="x" in ballE)
```
```   602   apply (erule_tac x="y" in ballE)
```
```   603    apply simp
```
```   604   apply (simp add: in_respects)
```
```   605  apply (simp add: in_respects)
```
```   606 apply (erule_tac exE)
```
```   607  apply rule
```
```   608  apply (rule_tac x="repf x" in exI)
```
```   609  apply (simp only: in_respects)
```
```   610   apply rule
```
```   611  apply (metis Quotient_rel_rep[OF a])
```
```   612 using a unfolding Quotient_def apply (simp)
```
```   613 apply rule+
```
```   614 using a unfolding Quotient_def in_respects
```
```   615 apply metis
```
```   616 done
```
```   617
```
```   618 lemma bex1_bexeq_reg:
```
```   619   shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
```
```   620   apply (simp add: Ex1_def Bex1_rel_def in_respects)
```
```   621   apply clarify
```
```   622   apply auto
```
```   623   apply (rule bexI)
```
```   624   apply assumption
```
```   625   apply (simp add: in_respects)
```
```   626   apply (simp add: in_respects)
```
```   627   apply auto
```
```   628   done
```
```   629
```
```   630 lemma bex1_bexeq_reg_eqv:
```
```   631   assumes a: "equivp R"
```
```   632   shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
```
```   633   using equivp_reflp[OF a]
```
```   634   apply (intro impI)
```
```   635   apply (elim ex1E)
```
```   636   apply (rule mp[OF bex1_bexeq_reg])
```
```   637   apply (rule_tac a="x" in ex1I)
```
```   638   apply (subst in_respects)
```
```   639   apply (rule conjI)
```
```   640   apply assumption
```
```   641   apply assumption
```
```   642   apply clarify
```
```   643   apply (erule_tac x="xa" in allE)
```
```   644   apply simp
```
```   645   done
```
```   646
```
```   647 subsection {* Various respects and preserve lemmas *}
```
```   648
```
```   649 lemma quot_rel_rsp:
```
```   650   assumes a: "Quotient R Abs Rep"
```
```   651   shows "(R ===> R ===> op =) R R"
```
```   652   apply(rule fun_relI)+
```
```   653   apply(rule equals_rsp[OF a])
```
```   654   apply(assumption)+
```
```   655   done
```
```   656
```
```   657 lemma o_prs:
```
```   658   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   659   and     q2: "Quotient R2 Abs2 Rep2"
```
```   660   and     q3: "Quotient R3 Abs3 Rep3"
```
```   661   shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
```
```   662   and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
```
```   663   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3]
```
```   664   by (simp_all add: fun_eq_iff)
```
```   665
```
```   666 lemma o_rsp:
```
```   667   "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
```
```   668   "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
```
```   669   by (auto intro!: fun_relI elim: fun_relE)
```
```   670
```
```   671 lemma cond_prs:
```
```   672   assumes a: "Quotient R absf repf"
```
```   673   shows "absf (if a then repf b else repf c) = (if a then b else c)"
```
```   674   using a unfolding Quotient_def by auto
```
```   675
```
```   676 lemma if_prs:
```
```   677   assumes q: "Quotient R Abs Rep"
```
```   678   shows "(id ---> Rep ---> Rep ---> Abs) If = If"
```
```   679   using Quotient_abs_rep[OF q]
```
```   680   by (auto simp add: fun_eq_iff)
```
```   681
```
```   682 lemma if_rsp:
```
```   683   assumes q: "Quotient R Abs Rep"
```
```   684   shows "(op = ===> R ===> R ===> R) If If"
```
```   685   by (auto intro!: fun_relI)
```
```   686
```
```   687 lemma let_prs:
```
```   688   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   689   and     q2: "Quotient R2 Abs2 Rep2"
```
```   690   shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
```
```   691   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
```
```   692   by (auto simp add: fun_eq_iff)
```
```   693
```
```   694 lemma let_rsp:
```
```   695   shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
```
```   696   by (auto intro!: fun_relI elim: fun_relE)
```
```   697
```
```   698 lemma mem_rsp:
```
```   699   shows "(R1 ===> (R1 ===> R2) ===> R2) op \<in> op \<in>"
```
```   700   by (auto intro!: fun_relI elim: fun_relE simp add: mem_def)
```
```   701
```
```   702 lemma mem_prs:
```
```   703   assumes a1: "Quotient R1 Abs1 Rep1"
```
```   704   and     a2: "Quotient R2 Abs2 Rep2"
```
```   705   shows "(Rep1 ---> (Abs1 ---> Rep2) ---> Abs2) op \<in> = op \<in>"
```
```   706   by (simp add: fun_eq_iff mem_def Quotient_abs_rep[OF a1] Quotient_abs_rep[OF a2])
```
```   707
```
```   708 lemma id_rsp:
```
```   709   shows "(R ===> R) id id"
```
```   710   by (auto intro: fun_relI)
```
```   711
```
```   712 lemma id_prs:
```
```   713   assumes a: "Quotient R Abs Rep"
```
```   714   shows "(Rep ---> Abs) id = id"
```
```   715   by (simp add: fun_eq_iff Quotient_abs_rep [OF a])
```
```   716
```
```   717
```
```   718 locale quot_type =
```
```   719   fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   720   and   Abs :: "('a \<Rightarrow> bool) \<Rightarrow> 'b"
```
```   721   and   Rep :: "'b \<Rightarrow> ('a \<Rightarrow> bool)"
```
```   722   assumes equivp: "part_equivp R"
```
```   723   and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = R x"
```
```   724   and     rep_inverse: "\<And>x. Abs (Rep x) = x"
```
```   725   and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = R x))) \<Longrightarrow> (Rep (Abs c)) = c"
```
```   726   and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
```
```   727 begin
```
```   728
```
```   729 definition
```
```   730   abs :: "'a \<Rightarrow> 'b"
```
```   731 where
```
```   732   "abs x = Abs (R x)"
```
```   733
```
```   734 definition
```
```   735   rep :: "'b \<Rightarrow> 'a"
```
```   736 where
```
```   737   "rep a = Eps (Rep a)"
```
```   738
```
```   739 lemma homeier5:
```
```   740   assumes a: "R r r"
```
```   741   shows "Rep (Abs (R r)) = R r"
```
```   742   apply (subst abs_inverse)
```
```   743   using a by auto
```
```   744
```
```   745 theorem homeier6:
```
```   746   assumes a: "R r r"
```
```   747   and b: "R s s"
```
```   748   shows "Abs (R r) = Abs (R s) \<longleftrightarrow> R r = R s"
```
```   749   by (metis a b homeier5)
```
```   750
```
```   751 theorem homeier8:
```
```   752   assumes "R r r"
```
```   753   shows "R (Eps (R r)) = R r"
```
```   754   using assms equivp[simplified part_equivp_def]
```
```   755   apply clarify
```
```   756   by (metis assms exE_some)
```
```   757
```
```   758 lemma Quotient:
```
```   759   shows "Quotient R abs rep"
```
```   760   unfolding Quotient_def abs_def rep_def
```
```   761   proof (intro conjI allI)
```
```   762     fix a r s
```
```   763     show "Abs (R (Eps (Rep a))) = a"
```
```   764       by (metis equivp exE_some part_equivp_def rep_inverse rep_prop)
```
```   765     show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (R r) = Abs (R s))"
```
```   766       by (metis homeier6 equivp[simplified part_equivp_def])
```
```   767     show "R (Eps (Rep a)) (Eps (Rep a))" proof -
```
```   768       obtain x where r: "R x x" and rep: "Rep a = R x" using rep_prop[of a] by auto
```
```   769       have "R (Eps (R x)) x" using homeier8 r by simp
```
```   770       then have "R x (Eps (R x))" using part_equivp_symp[OF equivp] by fast
```
```   771       then have "R (Eps (R x)) (Eps (R x))" using homeier8[OF r] by simp
```
```   772       then show "R (Eps (Rep a)) (Eps (Rep a))" using rep by simp
```
```   773     qed
```
```   774   qed
```
```   775
```
```   776 end
```
```   777
```
```   778
```
```   779 subsection {* ML setup *}
```
```   780
```
```   781 text {* Auxiliary data for the quotient package *}
```
```   782
```
```   783 use "Tools/Quotient/quotient_info.ML"
```
```   784
```
```   785 declare [[map "fun" = (map_fun, fun_rel)]]
```
```   786
```
```   787 lemmas [quot_thm] = fun_quotient
```
```   788 lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp mem_rsp id_rsp
```
```   789 lemmas [quot_preserve] = if_prs o_prs let_prs mem_prs id_prs
```
```   790 lemmas [quot_equiv] = identity_equivp
```
```   791
```
```   792
```
```   793 text {* Lemmas about simplifying id's. *}
```
```   794 lemmas [id_simps] =
```
```   795   id_def[symmetric]
```
```   796   map_fun_id
```
```   797   id_apply
```
```   798   id_o
```
```   799   o_id
```
```   800   eq_comp_r
```
```   801
```
```   802 text {* Translation functions for the lifting process. *}
```
```   803 use "Tools/Quotient/quotient_term.ML"
```
```   804
```
```   805
```
```   806 text {* Definitions of the quotient types. *}
```
```   807 use "Tools/Quotient/quotient_typ.ML"
```
```   808
```
```   809
```
```   810 text {* Definitions for quotient constants. *}
```
```   811 use "Tools/Quotient/quotient_def.ML"
```
```   812
```
```   813
```
```   814 text {*
```
```   815   An auxiliary constant for recording some information
```
```   816   about the lifted theorem in a tactic.
```
```   817 *}
```
```   818 definition
```
```   819   Quot_True :: "'a \<Rightarrow> bool"
```
```   820 where
```
```   821   "Quot_True x \<longleftrightarrow> True"
```
```   822
```
```   823 lemma
```
```   824   shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
```
```   825   and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
```
```   826   and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
```
```   827   and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
```
```   828   and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
```
```   829   by (simp_all add: Quot_True_def ext)
```
```   830
```
```   831 lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
```
```   832   by (simp add: Quot_True_def)
```
```   833
```
```   834
```
```   835 text {* Tactics for proving the lifted theorems *}
```
```   836 use "Tools/Quotient/quotient_tacs.ML"
```
```   837
```
```   838 subsection {* Methods / Interface *}
```
```   839
```
```   840 method_setup lifting =
```
```   841   {* Attrib.thms >> (fn thms => fn ctxt =>
```
```   842        SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_tac ctxt [] thms))) *}
```
```   843   {* lifts theorems to quotient types *}
```
```   844
```
```   845 method_setup lifting_setup =
```
```   846   {* Attrib.thm >> (fn thm => fn ctxt =>
```
```   847        SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_procedure_tac ctxt [] thm))) *}
```
```   848   {* sets up the three goals for the quotient lifting procedure *}
```
```   849
```
```   850 method_setup descending =
```
```   851   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.descend_tac ctxt []))) *}
```
```   852   {* decends theorems to the raw level *}
```
```   853
```
```   854 method_setup descending_setup =
```
```   855   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.descend_procedure_tac ctxt []))) *}
```
```   856   {* sets up the three goals for the decending theorems *}
```
```   857
```
```   858 method_setup regularize =
```
```   859   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.regularize_tac ctxt))) *}
```
```   860   {* proves the regularization goals from the quotient lifting procedure *}
```
```   861
```
```   862 method_setup injection =
```
```   863   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.all_injection_tac ctxt))) *}
```
```   864   {* proves the rep/abs injection goals from the quotient lifting procedure *}
```
```   865
```
```   866 method_setup cleaning =
```
```   867   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.clean_tac ctxt))) *}
```
```   868   {* proves the cleaning goals from the quotient lifting procedure *}
```
```   869
```
```   870 attribute_setup quot_lifted =
```
```   871   {* Scan.succeed Quotient_Tacs.lifted_attrib *}
```
```   872   {* lifts theorems to quotient types *}
```
```   873
```
```   874 no_notation
```
```   875   rel_conj (infixr "OOO" 75) and
```
```   876   map_fun (infixr "--->" 55) and
```
```   877   fun_rel (infixr "===>" 55)
```
```   878
```
```   879 end
```