src/HOL/Fun.thy
 author paulson Mon Feb 22 14:37:56 2016 +0000 (2016-02-22) changeset 62379 340738057c8c parent 61955 e96292f32c3c child 62390 842917225d56 permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
```     1 (*  Title:      HOL/Fun.thy
```
```     2     Author:     Tobias Nipkow, Cambridge University Computer Laboratory
```
```     3     Author:     Andrei Popescu, TU Muenchen
```
```     4     Copyright   1994, 2012
```
```     5 *)
```
```     6
```
```     7 section \<open>Notions about functions\<close>
```
```     8
```
```     9 theory Fun
```
```    10 imports Set
```
```    11 keywords "functor" :: thy_goal
```
```    12 begin
```
```    13
```
```    14 lemma apply_inverse:
```
```    15   "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"
```
```    16   by auto
```
```    17
```
```    18 text\<open>Uniqueness, so NOT the axiom of choice.\<close>
```
```    19 lemma uniq_choice: "\<forall>x. \<exists>!y. Q x y \<Longrightarrow> \<exists>f. \<forall>x. Q x (f x)"
```
```    20   by (force intro: theI')
```
```    21
```
```    22 lemma b_uniq_choice: "\<forall>x\<in>S. \<exists>!y. Q x y \<Longrightarrow> \<exists>f. \<forall>x\<in>S. Q x (f x)"
```
```    23   by (force intro: theI')
```
```    24
```
```    25 subsection \<open>The Identity Function \<open>id\<close>\<close>
```
```    26
```
```    27 definition id :: "'a \<Rightarrow> 'a" where
```
```    28   "id = (\<lambda>x. x)"
```
```    29
```
```    30 lemma id_apply [simp]: "id x = x"
```
```    31   by (simp add: id_def)
```
```    32
```
```    33 lemma image_id [simp]: "image id = id"
```
```    34   by (simp add: id_def fun_eq_iff)
```
```    35
```
```    36 lemma vimage_id [simp]: "vimage id = id"
```
```    37   by (simp add: id_def fun_eq_iff)
```
```    38
```
```    39 code_printing
```
```    40   constant id \<rightharpoonup> (Haskell) "id"
```
```    41
```
```    42
```
```    43 subsection \<open>The Composition Operator \<open>f \<circ> g\<close>\<close>
```
```    44
```
```    45 definition comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c"  (infixl "\<circ>" 55)
```
```    46   where "f \<circ> g = (\<lambda>x. f (g x))"
```
```    47
```
```    48 notation (ASCII)
```
```    49   comp  (infixl "o" 55)
```
```    50
```
```    51 lemma comp_apply [simp]: "(f o g) x = f (g x)"
```
```    52   by (simp add: comp_def)
```
```    53
```
```    54 lemma comp_assoc: "(f o g) o h = f o (g o h)"
```
```    55   by (simp add: fun_eq_iff)
```
```    56
```
```    57 lemma id_comp [simp]: "id o g = g"
```
```    58   by (simp add: fun_eq_iff)
```
```    59
```
```    60 lemma comp_id [simp]: "f o id = f"
```
```    61   by (simp add: fun_eq_iff)
```
```    62
```
```    63 lemma comp_eq_dest:
```
```    64   "a o b = c o d \<Longrightarrow> a (b v) = c (d v)"
```
```    65   by (simp add: fun_eq_iff)
```
```    66
```
```    67 lemma comp_eq_elim:
```
```    68   "a o b = c o d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R"
```
```    69   by (simp add: fun_eq_iff)
```
```    70
```
```    71 lemma comp_eq_dest_lhs: "a o b = c \<Longrightarrow> a (b v) = c v"
```
```    72   by clarsimp
```
```    73
```
```    74 lemma comp_eq_id_dest: "a o b = id o c \<Longrightarrow> a (b v) = c v"
```
```    75   by clarsimp
```
```    76
```
```    77 lemma image_comp:
```
```    78   "f ` (g ` r) = (f o g) ` r"
```
```    79   by auto
```
```    80
```
```    81 lemma vimage_comp:
```
```    82   "f -` (g -` x) = (g \<circ> f) -` x"
```
```    83   by auto
```
```    84
```
```    85 lemma image_eq_imp_comp: "f ` A = g ` B \<Longrightarrow> (h o f) ` A = (h o g) ` B"
```
```    86   by (auto simp: comp_def elim!: equalityE)
```
```    87
```
```    88 lemma image_bind: "f ` (Set.bind A g) = Set.bind A (op ` f \<circ> g)"
```
```    89 by(auto simp add: Set.bind_def)
```
```    90
```
```    91 lemma bind_image: "Set.bind (f ` A) g = Set.bind A (g \<circ> f)"
```
```    92 by(auto simp add: Set.bind_def)
```
```    93
```
```    94 lemma (in group_add) minus_comp_minus [simp]:
```
```    95   "uminus \<circ> uminus = id"
```
```    96   by (simp add: fun_eq_iff)
```
```    97
```
```    98 lemma (in boolean_algebra) minus_comp_minus [simp]:
```
```    99   "uminus \<circ> uminus = id"
```
```   100   by (simp add: fun_eq_iff)
```
```   101
```
```   102 code_printing
```
```   103   constant comp \<rightharpoonup> (SML) infixl 5 "o" and (Haskell) infixr 9 "."
```
```   104
```
```   105
```
```   106 subsection \<open>The Forward Composition Operator \<open>fcomp\<close>\<close>
```
```   107
```
```   108 definition fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>>" 60) where
```
```   109   "f \<circ>> g = (\<lambda>x. g (f x))"
```
```   110
```
```   111 lemma fcomp_apply [simp]:  "(f \<circ>> g) x = g (f x)"
```
```   112   by (simp add: fcomp_def)
```
```   113
```
```   114 lemma fcomp_assoc: "(f \<circ>> g) \<circ>> h = f \<circ>> (g \<circ>> h)"
```
```   115   by (simp add: fcomp_def)
```
```   116
```
```   117 lemma id_fcomp [simp]: "id \<circ>> g = g"
```
```   118   by (simp add: fcomp_def)
```
```   119
```
```   120 lemma fcomp_id [simp]: "f \<circ>> id = f"
```
```   121   by (simp add: fcomp_def)
```
```   122
```
```   123 lemma fcomp_comp: "fcomp f g = comp g f"
```
```   124   by (simp add: ext)
```
```   125
```
```   126 code_printing
```
```   127   constant fcomp \<rightharpoonup> (Eval) infixl 1 "#>"
```
```   128
```
```   129 no_notation fcomp (infixl "\<circ>>" 60)
```
```   130
```
```   131
```
```   132 subsection \<open>Mapping functions\<close>
```
```   133
```
```   134 definition map_fun :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'd" where
```
```   135   "map_fun f g h = g \<circ> h \<circ> f"
```
```   136
```
```   137 lemma map_fun_apply [simp]:
```
```   138   "map_fun f g h x = g (h (f x))"
```
```   139   by (simp add: map_fun_def)
```
```   140
```
```   141
```
```   142 subsection \<open>Injectivity and Bijectivity\<close>
```
```   143
```
```   144 definition inj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool" where \<comment> "injective"
```
```   145   "inj_on f A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
```
```   146
```
```   147 definition bij_betw :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" where \<comment> "bijective"
```
```   148   "bij_betw f A B \<longleftrightarrow> inj_on f A \<and> f ` A = B"
```
```   149
```
```   150 text\<open>A common special case: functions injective, surjective or bijective over
```
```   151 the entire domain type.\<close>
```
```   152
```
```   153 abbreviation
```
```   154   "inj f \<equiv> inj_on f UNIV"
```
```   155
```
```   156 abbreviation surj :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" where \<comment> "surjective"
```
```   157   "surj f \<equiv> (range f = UNIV)"
```
```   158
```
```   159 abbreviation
```
```   160   "bij f \<equiv> bij_betw f UNIV UNIV"
```
```   161
```
```   162 text\<open>The negated case:\<close>
```
```   163 translations
```
```   164 "\<not> CONST surj f" <= "CONST range f \<noteq> CONST UNIV"
```
```   165
```
```   166 lemma injI:
```
```   167   assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
```
```   168   shows "inj f"
```
```   169   using assms unfolding inj_on_def by auto
```
```   170
```
```   171 theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"
```
```   172   by (unfold inj_on_def, blast)
```
```   173
```
```   174 lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
```
```   175 by (simp add: inj_on_def)
```
```   176
```
```   177 lemma inj_on_eq_iff: "\<lbrakk>inj_on f A; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> (f x = f y) = (x = y)"
```
```   178 by (force simp add: inj_on_def)
```
```   179
```
```   180 lemma inj_on_cong:
```
```   181   "(\<And> a. a : A \<Longrightarrow> f a = g a) \<Longrightarrow> inj_on f A = inj_on g A"
```
```   182 unfolding inj_on_def by auto
```
```   183
```
```   184 lemma inj_on_strict_subset:
```
```   185   "inj_on f B \<Longrightarrow> A \<subset> B \<Longrightarrow> f ` A \<subset> f ` B"
```
```   186   unfolding inj_on_def by blast
```
```   187
```
```   188 lemma inj_comp:
```
```   189   "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)"
```
```   190   by (simp add: inj_on_def)
```
```   191
```
```   192 lemma inj_fun: "inj f \<Longrightarrow> inj (\<lambda>x y. f x)"
```
```   193   by (simp add: inj_on_def fun_eq_iff)
```
```   194
```
```   195 lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)"
```
```   196 by (simp add: inj_on_eq_iff)
```
```   197
```
```   198 lemma inj_on_id[simp]: "inj_on id A"
```
```   199   by (simp add: inj_on_def)
```
```   200
```
```   201 lemma inj_on_id2[simp]: "inj_on (%x. x) A"
```
```   202 by (simp add: inj_on_def)
```
```   203
```
```   204 lemma inj_on_Int: "inj_on f A \<or> inj_on f B \<Longrightarrow> inj_on f (A \<inter> B)"
```
```   205 unfolding inj_on_def by blast
```
```   206
```
```   207 lemma surj_id: "surj id"
```
```   208 by simp
```
```   209
```
```   210 lemma bij_id[simp]: "bij id"
```
```   211 by (simp add: bij_betw_def)
```
```   212
```
```   213 lemma inj_onI:
```
```   214     "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
```
```   215 by (simp add: inj_on_def)
```
```   216
```
```   217 lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
```
```   218 by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
```
```   219
```
```   220 lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
```
```   221 by (unfold inj_on_def, blast)
```
```   222
```
```   223 lemma comp_inj_on:
```
```   224      "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
```
```   225 by (simp add: comp_def inj_on_def)
```
```   226
```
```   227 lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
```
```   228   by (simp add: inj_on_def) blast
```
```   229
```
```   230 lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
```
```   231   inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
```
```   232 apply(unfold inj_on_def)
```
```   233 apply blast
```
```   234 done
```
```   235
```
```   236 lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
```
```   237 by (unfold inj_on_def, blast)
```
```   238
```
```   239 lemma inj_singleton: "inj (%s. {s})"
```
```   240 by (simp add: inj_on_def)
```
```   241
```
```   242 lemma inj_on_empty[iff]: "inj_on f {}"
```
```   243 by(simp add: inj_on_def)
```
```   244
```
```   245 lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
```
```   246 by (unfold inj_on_def, blast)
```
```   247
```
```   248 lemma inj_on_Un:
```
```   249  "inj_on f (A Un B) =
```
```   250   (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
```
```   251 apply(unfold inj_on_def)
```
```   252 apply (blast intro:sym)
```
```   253 done
```
```   254
```
```   255 lemma inj_on_insert[iff]:
```
```   256   "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
```
```   257 apply(unfold inj_on_def)
```
```   258 apply (blast intro:sym)
```
```   259 done
```
```   260
```
```   261 lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
```
```   262 apply(unfold inj_on_def)
```
```   263 apply (blast)
```
```   264 done
```
```   265
```
```   266 lemma comp_inj_on_iff:
```
```   267   "inj_on f A \<Longrightarrow> inj_on f' (f ` A) \<longleftrightarrow> inj_on (f' o f) A"
```
```   268 by(auto simp add: comp_inj_on inj_on_def)
```
```   269
```
```   270 lemma inj_on_imageI2:
```
```   271   "inj_on (f' o f) A \<Longrightarrow> inj_on f A"
```
```   272 by(auto simp add: comp_inj_on inj_on_def)
```
```   273
```
```   274 lemma inj_img_insertE:
```
```   275   assumes "inj_on f A"
```
```   276   assumes "x \<notin> B" and "insert x B = f ` A"
```
```   277   obtains x' A' where "x' \<notin> A'" and "A = insert x' A'"
```
```   278     and "x = f x'" and "B = f ` A'"
```
```   279 proof -
```
```   280   from assms have "x \<in> f ` A" by auto
```
```   281   then obtain x' where *: "x' \<in> A" "x = f x'" by auto
```
```   282   then have "A = insert x' (A - {x'})" by auto
```
```   283   with assms * have "B = f ` (A - {x'})"
```
```   284     by (auto dest: inj_on_contraD)
```
```   285   have "x' \<notin> A - {x'}" by simp
```
```   286   from \<open>x' \<notin> A - {x'}\<close> \<open>A = insert x' (A - {x'})\<close> \<open>x = f x'\<close> \<open>B = image f (A - {x'})\<close>
```
```   287   show ?thesis ..
```
```   288 qed
```
```   289
```
```   290 lemma linorder_injI:
```
```   291   assumes hyp: "\<And>x y. x < (y::'a::linorder) \<Longrightarrow> f x \<noteq> f y"
```
```   292   shows "inj f"
```
```   293   \<comment> \<open>Courtesy of Stephan Merz\<close>
```
```   294 proof (rule inj_onI)
```
```   295   fix x y
```
```   296   assume f_eq: "f x = f y"
```
```   297   show "x = y" by (rule linorder_cases) (auto dest: hyp simp: f_eq)
```
```   298 qed
```
```   299
```
```   300 lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)"
```
```   301   by auto
```
```   302
```
```   303 lemma surjI: assumes *: "\<And> x. g (f x) = x" shows "surj g"
```
```   304   using *[symmetric] by auto
```
```   305
```
```   306 lemma surjD: "surj f \<Longrightarrow> \<exists>x. y = f x"
```
```   307   by (simp add: surj_def)
```
```   308
```
```   309 lemma surjE: "surj f \<Longrightarrow> (\<And>x. y = f x \<Longrightarrow> C) \<Longrightarrow> C"
```
```   310   by (simp add: surj_def, blast)
```
```   311
```
```   312 lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
```
```   313 apply (simp add: comp_def surj_def, clarify)
```
```   314 apply (drule_tac x = y in spec, clarify)
```
```   315 apply (drule_tac x = x in spec, blast)
```
```   316 done
```
```   317
```
```   318 lemma bij_betw_imageI:
```
```   319   "\<lbrakk> inj_on f A; f ` A = B \<rbrakk> \<Longrightarrow> bij_betw f A B"
```
```   320 unfolding bij_betw_def by clarify
```
```   321
```
```   322 lemma bij_betw_imp_surj_on: "bij_betw f A B \<Longrightarrow> f ` A = B"
```
```   323   unfolding bij_betw_def by clarify
```
```   324
```
```   325 lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f"
```
```   326   unfolding bij_betw_def by auto
```
```   327
```
```   328 lemma bij_betw_empty1:
```
```   329   assumes "bij_betw f {} A"
```
```   330   shows "A = {}"
```
```   331 using assms unfolding bij_betw_def by blast
```
```   332
```
```   333 lemma bij_betw_empty2:
```
```   334   assumes "bij_betw f A {}"
```
```   335   shows "A = {}"
```
```   336 using assms unfolding bij_betw_def by blast
```
```   337
```
```   338 lemma inj_on_imp_bij_betw:
```
```   339   "inj_on f A \<Longrightarrow> bij_betw f A (f ` A)"
```
```   340 unfolding bij_betw_def by simp
```
```   341
```
```   342 lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f"
```
```   343   unfolding bij_betw_def ..
```
```   344
```
```   345 lemma bijI: "[| inj f; surj f |] ==> bij f"
```
```   346 by (simp add: bij_def)
```
```   347
```
```   348 lemma bij_is_inj: "bij f ==> inj f"
```
```   349 by (simp add: bij_def)
```
```   350
```
```   351 lemma bij_is_surj: "bij f ==> surj f"
```
```   352 by (simp add: bij_def)
```
```   353
```
```   354 lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
```
```   355 by (simp add: bij_betw_def)
```
```   356
```
```   357 lemma bij_betw_trans:
```
```   358   "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
```
```   359 by(auto simp add:bij_betw_def comp_inj_on)
```
```   360
```
```   361 lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
```
```   362   by (rule bij_betw_trans)
```
```   363
```
```   364 lemma bij_betw_comp_iff:
```
```   365   "bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' o f) A A''"
```
```   366 by(auto simp add: bij_betw_def inj_on_def)
```
```   367
```
```   368 lemma bij_betw_comp_iff2:
```
```   369   assumes BIJ: "bij_betw f' A' A''" and IM: "f ` A \<le> A'"
```
```   370   shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' o f) A A''"
```
```   371 using assms
```
```   372 proof(auto simp add: bij_betw_comp_iff)
```
```   373   assume *: "bij_betw (f' \<circ> f) A A''"
```
```   374   thus "bij_betw f A A'"
```
```   375   using IM
```
```   376   proof(auto simp add: bij_betw_def)
```
```   377     assume "inj_on (f' \<circ> f) A"
```
```   378     thus "inj_on f A" using inj_on_imageI2 by blast
```
```   379   next
```
```   380     fix a' assume **: "a' \<in> A'"
```
```   381     hence "f' a' \<in> A''" using BIJ unfolding bij_betw_def by auto
```
```   382     then obtain a where 1: "a \<in> A \<and> f'(f a) = f' a'" using *
```
```   383     unfolding bij_betw_def by force
```
```   384     hence "f a \<in> A'" using IM by auto
```
```   385     hence "f a = a'" using BIJ ** 1 unfolding bij_betw_def inj_on_def by auto
```
```   386     thus "a' \<in> f ` A" using 1 by auto
```
```   387   qed
```
```   388 qed
```
```   389
```
```   390 lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
```
```   391 proof -
```
```   392   have i: "inj_on f A" and s: "f ` A = B"
```
```   393     using assms by(auto simp:bij_betw_def)
```
```   394   let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
```
```   395   { fix a b assume P: "?P b a"
```
```   396     hence ex1: "\<exists>a. ?P b a" using s by blast
```
```   397     hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
```
```   398     hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
```
```   399   } note g = this
```
```   400   have "inj_on ?g B"
```
```   401   proof(rule inj_onI)
```
```   402     fix x y assume "x:B" "y:B" "?g x = ?g y"
```
```   403     from s \<open>x:B\<close> obtain a1 where a1: "?P x a1" by blast
```
```   404     from s \<open>y:B\<close> obtain a2 where a2: "?P y a2" by blast
```
```   405     from g[OF a1] a1 g[OF a2] a2 \<open>?g x = ?g y\<close> show "x=y" by simp
```
```   406   qed
```
```   407   moreover have "?g ` B = A"
```
```   408   proof(auto simp: image_def)
```
```   409     fix b assume "b:B"
```
```   410     with s obtain a where P: "?P b a" by blast
```
```   411     thus "?g b \<in> A" using g[OF P] by auto
```
```   412   next
```
```   413     fix a assume "a:A"
```
```   414     then obtain b where P: "?P b a" using s by blast
```
```   415     then have "b:B" using s by blast
```
```   416     with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
```
```   417   qed
```
```   418   ultimately show ?thesis by(auto simp:bij_betw_def)
```
```   419 qed
```
```   420
```
```   421 lemma bij_betw_cong:
```
```   422   "(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"
```
```   423 unfolding bij_betw_def inj_on_def by force
```
```   424
```
```   425 lemma bij_betw_id[intro, simp]:
```
```   426   "bij_betw id A A"
```
```   427 unfolding bij_betw_def id_def by auto
```
```   428
```
```   429 lemma bij_betw_id_iff:
```
```   430   "bij_betw id A B \<longleftrightarrow> A = B"
```
```   431 by(auto simp add: bij_betw_def)
```
```   432
```
```   433 lemma bij_betw_combine:
```
```   434   assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}"
```
```   435   shows "bij_betw f (A \<union> C) (B \<union> D)"
```
```   436   using assms unfolding bij_betw_def inj_on_Un image_Un by auto
```
```   437
```
```   438 lemma bij_betw_subset:
```
```   439   assumes BIJ: "bij_betw f A A'" and
```
```   440           SUB: "B \<le> A" and IM: "f ` B = B'"
```
```   441   shows "bij_betw f B B'"
```
```   442 using assms
```
```   443 by(unfold bij_betw_def inj_on_def, auto simp add: inj_on_def)
```
```   444
```
```   445 lemma bij_pointE:
```
```   446   assumes "bij f"
```
```   447   obtains x where "y = f x" and "\<And>x'. y = f x' \<Longrightarrow> x' = x"
```
```   448 proof -
```
```   449   from assms have "inj f" by (rule bij_is_inj)
```
```   450   moreover from assms have "surj f" by (rule bij_is_surj)
```
```   451   then have "y \<in> range f" by simp
```
```   452   ultimately have "\<exists>!x. y = f x" by (simp add: range_ex1_eq)
```
```   453   with that show thesis by blast
```
```   454 qed
```
```   455
```
```   456 lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
```
```   457 by simp
```
```   458
```
```   459 lemma surj_vimage_empty:
```
```   460   assumes "surj f" shows "f -` A = {} \<longleftrightarrow> A = {}"
```
```   461   using surj_image_vimage_eq[OF \<open>surj f\<close>, of A]
```
```   462   by (intro iffI) fastforce+
```
```   463
```
```   464 lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
```
```   465 by (simp add: inj_on_def, blast)
```
```   466
```
```   467 lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
```
```   468 by (blast intro: sym)
```
```   469
```
```   470 lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
```
```   471 by (unfold inj_on_def, blast)
```
```   472
```
```   473 lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
```
```   474 apply (unfold bij_def)
```
```   475 apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
```
```   476 done
```
```   477
```
```   478 lemma inj_on_image_eq_iff: "\<lbrakk> inj_on f C; A \<subseteq> C; B \<subseteq> C \<rbrakk> \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
```
```   479 by(fastforce simp add: inj_on_def)
```
```   480
```
```   481 lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
```
```   482 by(erule inj_on_image_eq_iff) simp_all
```
```   483
```
```   484 lemma inj_on_image_Int:
```
```   485    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
```
```   486   by (simp add: inj_on_def, blast)
```
```   487
```
```   488 lemma inj_on_image_set_diff:
```
```   489    "[| inj_on f C;  A-B \<subseteq> C;  B \<subseteq> C |] ==> f`(A-B) = f`A - f`B"
```
```   490   by (simp add: inj_on_def, blast)
```
```   491
```
```   492 lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
```
```   493   by (simp add: inj_on_def, blast)
```
```   494
```
```   495 lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
```
```   496 by (simp add: inj_on_def, blast)
```
```   497
```
```   498 lemma inj_on_image_mem_iff: "\<lbrakk>inj_on f B; a \<in> B; A \<subseteq> B\<rbrakk> \<Longrightarrow> f a \<in> f`A \<longleftrightarrow> a \<in> A"
```
```   499   by (auto simp: inj_on_def)
```
```   500
```
```   501 (*FIXME DELETE*)
```
```   502 lemma inj_on_image_mem_iff_alt: "inj_on f B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f a \<in> f`A \<Longrightarrow> a \<in> B \<Longrightarrow> a \<in> A"
```
```   503   by (blast dest: inj_onD)
```
```   504
```
```   505 lemma inj_image_mem_iff: "inj f \<Longrightarrow> f a \<in> f`A \<longleftrightarrow> a \<in> A"
```
```   506   by (blast dest: injD)
```
```   507
```
```   508 lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
```
```   509   by (blast dest: injD)
```
```   510
```
```   511 lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
```
```   512   by (blast dest: injD)
```
```   513
```
```   514 lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
```
```   515 by auto
```
```   516
```
```   517 lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
```
```   518 by (auto simp add: inj_on_def)
```
```   519
```
```   520 lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
```
```   521 apply (simp add: bij_def)
```
```   522 apply (rule equalityI)
```
```   523 apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
```
```   524 done
```
```   525
```
```   526 lemma inj_vimage_singleton: "inj f \<Longrightarrow> f -` {a} \<subseteq> {THE x. f x = a}"
```
```   527   \<comment> \<open>The inverse image of a singleton under an injective function
```
```   528          is included in a singleton.\<close>
```
```   529   apply (auto simp add: inj_on_def)
```
```   530   apply (blast intro: the_equality [symmetric])
```
```   531   done
```
```   532
```
```   533 lemma inj_on_vimage_singleton:
```
```   534   "inj_on f A \<Longrightarrow> f -` {a} \<inter> A \<subseteq> {THE x. x \<in> A \<and> f x = a}"
```
```   535   by (auto simp add: inj_on_def intro: the_equality [symmetric])
```
```   536
```
```   537 lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"
```
```   538   by (auto intro!: inj_onI)
```
```   539
```
```   540 lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"
```
```   541   by (auto intro!: inj_onI dest: strict_mono_eq)
```
```   542
```
```   543 lemma bij_betw_byWitness:
```
```   544 assumes LEFT: "\<forall>a \<in> A. f'(f a) = a" and
```
```   545         RIGHT: "\<forall>a' \<in> A'. f(f' a') = a'" and
```
```   546         IM1: "f ` A \<le> A'" and IM2: "f' ` A' \<le> A"
```
```   547 shows "bij_betw f A A'"
```
```   548 using assms
```
```   549 proof(unfold bij_betw_def inj_on_def, safe)
```
```   550   fix a b assume *: "a \<in> A" "b \<in> A" and **: "f a = f b"
```
```   551   have "a = f'(f a) \<and> b = f'(f b)" using * LEFT by simp
```
```   552   with ** show "a = b" by simp
```
```   553 next
```
```   554   fix a' assume *: "a' \<in> A'"
```
```   555   hence "f' a' \<in> A" using IM2 by blast
```
```   556   moreover
```
```   557   have "a' = f(f' a')" using * RIGHT by simp
```
```   558   ultimately show "a' \<in> f ` A" by blast
```
```   559 qed
```
```   560
```
```   561 corollary notIn_Un_bij_betw:
```
```   562 assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'" and
```
```   563        BIJ: "bij_betw f A A'"
```
```   564 shows "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
```
```   565 proof-
```
```   566   have "bij_betw f {b} {f b}"
```
```   567   unfolding bij_betw_def inj_on_def by simp
```
```   568   with assms show ?thesis
```
```   569   using bij_betw_combine[of f A A' "{b}" "{f b}"] by blast
```
```   570 qed
```
```   571
```
```   572 lemma notIn_Un_bij_betw3:
```
```   573 assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'"
```
```   574 shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})"
```
```   575 proof
```
```   576   assume "bij_betw f A A'"
```
```   577   thus "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
```
```   578   using assms notIn_Un_bij_betw[of b A f A'] by blast
```
```   579 next
```
```   580   assume *: "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
```
```   581   have "f ` A = A'"
```
```   582   proof(auto)
```
```   583     fix a assume **: "a \<in> A"
```
```   584     hence "f a \<in> A' \<union> {f b}" using * unfolding bij_betw_def by blast
```
```   585     moreover
```
```   586     {assume "f a = f b"
```
```   587      hence "a = b" using * ** unfolding bij_betw_def inj_on_def by blast
```
```   588      with NIN ** have False by blast
```
```   589     }
```
```   590     ultimately show "f a \<in> A'" by blast
```
```   591   next
```
```   592     fix a' assume **: "a' \<in> A'"
```
```   593     hence "a' \<in> f`(A \<union> {b})"
```
```   594     using * by (auto simp add: bij_betw_def)
```
```   595     then obtain a where 1: "a \<in> A \<union> {b} \<and> f a = a'" by blast
```
```   596     moreover
```
```   597     {assume "a = b" with 1 ** NIN' have False by blast
```
```   598     }
```
```   599     ultimately have "a \<in> A" by blast
```
```   600     with 1 show "a' \<in> f ` A" by blast
```
```   601   qed
```
```   602   thus "bij_betw f A A'" using * bij_betw_subset[of f "A \<union> {b}" _ A] by blast
```
```   603 qed
```
```   604
```
```   605
```
```   606 subsection\<open>Function Updating\<close>
```
```   607
```
```   608 definition fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
```
```   609   "fun_upd f a b == % x. if x=a then b else f x"
```
```   610
```
```   611 nonterminal updbinds and updbind
```
```   612
```
```   613 syntax
```
```   614   "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
```
```   615   ""         :: "updbind => updbinds"             ("_")
```
```   616   "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
```
```   617   "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000, 0] 900)
```
```   618
```
```   619 translations
```
```   620   "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
```
```   621   "f(x:=y)" == "CONST fun_upd f x y"
```
```   622
```
```   623 (* Hint: to define the sum of two functions (or maps), use case_sum.
```
```   624          A nice infix syntax could be defined by
```
```   625 notation
```
```   626   case_sum  (infixr "'(+')"80)
```
```   627 *)
```
```   628
```
```   629 lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
```
```   630 apply (simp add: fun_upd_def, safe)
```
```   631 apply (erule subst)
```
```   632 apply (rule_tac [2] ext, auto)
```
```   633 done
```
```   634
```
```   635 lemma fun_upd_idem: "f x = y ==> f(x:=y) = f"
```
```   636   by (simp only: fun_upd_idem_iff)
```
```   637
```
```   638 lemma fun_upd_triv [iff]: "f(x := f x) = f"
```
```   639   by (simp only: fun_upd_idem)
```
```   640
```
```   641 lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
```
```   642 by (simp add: fun_upd_def)
```
```   643
```
```   644 (* fun_upd_apply supersedes these two,   but they are useful
```
```   645    if fun_upd_apply is intentionally removed from the simpset *)
```
```   646 lemma fun_upd_same: "(f(x:=y)) x = y"
```
```   647 by simp
```
```   648
```
```   649 lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
```
```   650 by simp
```
```   651
```
```   652 lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
```
```   653 by (simp add: fun_eq_iff)
```
```   654
```
```   655 lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
```
```   656 by (rule ext, auto)
```
```   657
```
```   658 lemma inj_on_fun_updI:
```
```   659   "inj_on f A \<Longrightarrow> y \<notin> f ` A \<Longrightarrow> inj_on (f(x := y)) A"
```
```   660   by (fastforce simp: inj_on_def)
```
```   661
```
```   662 lemma fun_upd_image:
```
```   663      "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
```
```   664 by auto
```
```   665
```
```   666 lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
```
```   667   by auto
```
```   668
```
```   669 lemma fun_upd_eqD: "f(x := y) = g(x := z) \<Longrightarrow> y = z"
```
```   670 by(simp add: fun_eq_iff split: split_if_asm)
```
```   671
```
```   672 subsection \<open>\<open>override_on\<close>\<close>
```
```   673
```
```   674 definition override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" where
```
```   675   "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
```
```   676
```
```   677 lemma override_on_emptyset[simp]: "override_on f g {} = f"
```
```   678 by(simp add:override_on_def)
```
```   679
```
```   680 lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
```
```   681 by(simp add:override_on_def)
```
```   682
```
```   683 lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
```
```   684 by(simp add:override_on_def)
```
```   685
```
```   686
```
```   687 subsection \<open>\<open>swap\<close>\<close>
```
```   688
```
```   689 definition swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
```
```   690 where
```
```   691   "swap a b f = f (a := f b, b:= f a)"
```
```   692
```
```   693 lemma swap_apply [simp]:
```
```   694   "swap a b f a = f b"
```
```   695   "swap a b f b = f a"
```
```   696   "c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> swap a b f c = f c"
```
```   697   by (simp_all add: swap_def)
```
```   698
```
```   699 lemma swap_self [simp]:
```
```   700   "swap a a f = f"
```
```   701   by (simp add: swap_def)
```
```   702
```
```   703 lemma swap_commute:
```
```   704   "swap a b f = swap b a f"
```
```   705   by (simp add: fun_upd_def swap_def fun_eq_iff)
```
```   706
```
```   707 lemma swap_nilpotent [simp]:
```
```   708   "swap a b (swap a b f) = f"
```
```   709   by (rule ext, simp add: fun_upd_def swap_def)
```
```   710
```
```   711 lemma swap_comp_involutory [simp]:
```
```   712   "swap a b \<circ> swap a b = id"
```
```   713   by (rule ext) simp
```
```   714
```
```   715 lemma swap_triple:
```
```   716   assumes "a \<noteq> c" and "b \<noteq> c"
```
```   717   shows "swap a b (swap b c (swap a b f)) = swap a c f"
```
```   718   using assms by (simp add: fun_eq_iff swap_def)
```
```   719
```
```   720 lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
```
```   721   by (rule ext, simp add: fun_upd_def swap_def)
```
```   722
```
```   723 lemma swap_image_eq [simp]:
```
```   724   assumes "a \<in> A" "b \<in> A" shows "swap a b f ` A = f ` A"
```
```   725 proof -
```
```   726   have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A"
```
```   727     using assms by (auto simp: image_iff swap_def)
```
```   728   then have "swap a b (swap a b f) ` A \<subseteq> (swap a b f) ` A" .
```
```   729   with subset[of f] show ?thesis by auto
```
```   730 qed
```
```   731
```
```   732 lemma inj_on_imp_inj_on_swap:
```
```   733   "\<lbrakk>inj_on f A; a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> inj_on (swap a b f) A"
```
```   734   by (simp add: inj_on_def swap_def, blast)
```
```   735
```
```   736 lemma inj_on_swap_iff [simp]:
```
```   737   assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A"
```
```   738 proof
```
```   739   assume "inj_on (swap a b f) A"
```
```   740   with A have "inj_on (swap a b (swap a b f)) A"
```
```   741     by (iprover intro: inj_on_imp_inj_on_swap)
```
```   742   thus "inj_on f A" by simp
```
```   743 next
```
```   744   assume "inj_on f A"
```
```   745   with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
```
```   746 qed
```
```   747
```
```   748 lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)"
```
```   749   by simp
```
```   750
```
```   751 lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f"
```
```   752   by simp
```
```   753
```
```   754 lemma bij_betw_swap_iff [simp]:
```
```   755   "\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B"
```
```   756   by (auto simp: bij_betw_def)
```
```   757
```
```   758 lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f"
```
```   759   by simp
```
```   760
```
```   761 hide_const (open) swap
```
```   762
```
```   763
```
```   764 subsection \<open>Inversion of injective functions\<close>
```
```   765
```
```   766 definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
```
```   767   "the_inv_into A f == %x. THE y. y : A & f y = x"
```
```   768
```
```   769 lemma the_inv_into_f_f:
```
```   770   "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
```
```   771 apply (simp add: the_inv_into_def inj_on_def)
```
```   772 apply blast
```
```   773 done
```
```   774
```
```   775 lemma f_the_inv_into_f:
```
```   776   "inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"
```
```   777 apply (simp add: the_inv_into_def)
```
```   778 apply (rule the1I2)
```
```   779  apply(blast dest: inj_onD)
```
```   780 apply blast
```
```   781 done
```
```   782
```
```   783 lemma the_inv_into_into:
```
```   784   "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
```
```   785 apply (simp add: the_inv_into_def)
```
```   786 apply (rule the1I2)
```
```   787  apply(blast dest: inj_onD)
```
```   788 apply blast
```
```   789 done
```
```   790
```
```   791 lemma the_inv_into_onto[simp]:
```
```   792   "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
```
```   793 by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
```
```   794
```
```   795 lemma the_inv_into_f_eq:
```
```   796   "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
```
```   797   apply (erule subst)
```
```   798   apply (erule the_inv_into_f_f, assumption)
```
```   799   done
```
```   800
```
```   801 lemma the_inv_into_comp:
```
```   802   "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
```
```   803   the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
```
```   804 apply (rule the_inv_into_f_eq)
```
```   805   apply (fast intro: comp_inj_on)
```
```   806  apply (simp add: f_the_inv_into_f the_inv_into_into)
```
```   807 apply (simp add: the_inv_into_into)
```
```   808 done
```
```   809
```
```   810 lemma inj_on_the_inv_into:
```
```   811   "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
```
```   812 by (auto intro: inj_onI simp: the_inv_into_f_f)
```
```   813
```
```   814 lemma bij_betw_the_inv_into:
```
```   815   "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
```
```   816 by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
```
```   817
```
```   818 abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
```
```   819   "the_inv f \<equiv> the_inv_into UNIV f"
```
```   820
```
```   821 lemma the_inv_f_f:
```
```   822   assumes "inj f"
```
```   823   shows "the_inv f (f x) = x" using assms UNIV_I
```
```   824   by (rule the_inv_into_f_f)
```
```   825
```
```   826
```
```   827 subsection \<open>Cantor's Paradox\<close>
```
```   828
```
```   829 lemma Cantors_paradox:
```
```   830   "\<not>(\<exists>f. f ` A = Pow A)"
```
```   831 proof clarify
```
```   832   fix f assume "f ` A = Pow A" hence *: "Pow A \<le> f ` A" by blast
```
```   833   let ?X = "{a \<in> A. a \<notin> f a}"
```
```   834   have "?X \<in> Pow A" unfolding Pow_def by auto
```
```   835   with * obtain x where "x \<in> A \<and> f x = ?X" by blast
```
```   836   thus False by best
```
```   837 qed
```
```   838
```
```   839 subsection \<open>Setup\<close>
```
```   840
```
```   841 subsubsection \<open>Proof tools\<close>
```
```   842
```
```   843 text \<open>simplifies terms of the form
```
```   844   f(...,x:=y,...,x:=z,...) to f(...,x:=z,...)\<close>
```
```   845
```
```   846 simproc_setup fun_upd2 ("f(v := w, x := y)") = \<open>fn _ =>
```
```   847 let
```
```   848   fun gen_fun_upd NONE T _ _ = NONE
```
```   849     | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) \$ f \$ x \$ y)
```
```   850   fun dest_fun_T1 (Type (_, T :: Ts)) = T
```
```   851   fun find_double (t as Const (@{const_name fun_upd},T) \$ f \$ x \$ y) =
```
```   852     let
```
```   853       fun find (Const (@{const_name fun_upd},T) \$ g \$ v \$ w) =
```
```   854             if v aconv x then SOME g else gen_fun_upd (find g) T v w
```
```   855         | find t = NONE
```
```   856     in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
```
```   857
```
```   858   val ss = simpset_of @{context}
```
```   859
```
```   860   fun proc ctxt ct =
```
```   861     let
```
```   862       val t = Thm.term_of ct
```
```   863     in
```
```   864       case find_double t of
```
```   865         (T, NONE) => NONE
```
```   866       | (T, SOME rhs) =>
```
```   867           SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
```
```   868             (fn _ =>
```
```   869               resolve_tac ctxt [eq_reflection] 1 THEN
```
```   870               resolve_tac ctxt @{thms ext} 1 THEN
```
```   871               simp_tac (put_simpset ss ctxt) 1))
```
```   872     end
```
```   873 in proc end
```
```   874 \<close>
```
```   875
```
```   876
```
```   877 subsubsection \<open>Functorial structure of types\<close>
```
```   878
```
```   879 ML_file "Tools/functor.ML"
```
```   880
```
```   881 functor map_fun: map_fun
```
```   882   by (simp_all add: fun_eq_iff)
```
```   883
```
```   884 functor vimage
```
```   885   by (simp_all add: fun_eq_iff vimage_comp)
```
```   886
```
```   887 text \<open>Legacy theorem names\<close>
```
```   888
```
```   889 lemmas o_def = comp_def
```
```   890 lemmas o_apply = comp_apply
```
```   891 lemmas o_assoc = comp_assoc [symmetric]
```
```   892 lemmas id_o = id_comp
```
```   893 lemmas o_id = comp_id
```
```   894 lemmas o_eq_dest = comp_eq_dest
```
```   895 lemmas o_eq_elim = comp_eq_elim
```
```   896 lemmas o_eq_dest_lhs = comp_eq_dest_lhs
```
```   897 lemmas o_eq_id_dest = comp_eq_id_dest
```
```   898
```
```   899 end
```
```   900
```