src/HOL/Library/FuncSet.thy
 author haftmann Fri Mar 22 19:18:08 2019 +0000 (3 months ago) changeset 69946 494934c30f38 parent 69939 812ce526da33 child 70063 adaa0a6ea4fe permissions -rw-r--r--
improved code equations taken over from AFP
```     1 (*  Title:      HOL/Library/FuncSet.thy
```
```     2     Author:     Florian Kammueller and Lawrence C Paulson, Lukas Bulwahn
```
```     3 *)
```
```     4
```
```     5 section \<open>Pi and Function Sets\<close>
```
```     6
```
```     7 theory FuncSet
```
```     8   imports Main
```
```     9   abbrevs PiE = "Pi\<^sub>E"
```
```    10     and PIE = "\<Pi>\<^sub>E"
```
```    11 begin
```
```    12
```
```    13 definition Pi :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set"
```
```    14   where "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}"
```
```    15
```
```    16 definition extensional :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) set"
```
```    17   where "extensional A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = undefined}"
```
```    18
```
```    19 definition "restrict" :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
```
```    20   where "restrict f A = (\<lambda>x. if x \<in> A then f x else undefined)"
```
```    21
```
```    22 abbreviation funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  (infixr "\<rightarrow>" 60)
```
```    23   where "A \<rightarrow> B \<equiv> Pi A (\<lambda>_. B)"
```
```    24
```
```    25 syntax
```
```    26   "_Pi" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
```
```    27   "_lam" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
```
```    28 translations
```
```    29   "\<Pi> x\<in>A. B" \<rightleftharpoons> "CONST Pi A (\<lambda>x. B)"
```
```    30   "\<lambda>x\<in>A. f" \<rightleftharpoons> "CONST restrict (\<lambda>x. f) A"
```
```    31
```
```    32 definition "compose" :: "'a set \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'c)"
```
```    33   where "compose A g f = (\<lambda>x\<in>A. g (f x))"
```
```    34
```
```    35
```
```    36 subsection \<open>Basic Properties of \<^term>\<open>Pi\<close>\<close>
```
```    37
```
```    38 lemma Pi_I[intro!]: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi A B"
```
```    39   by (simp add: Pi_def)
```
```    40
```
```    41 lemma Pi_I'[simp]: "(\<And>x. x \<in> A \<longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi A B"
```
```    42   by (simp add:Pi_def)
```
```    43
```
```    44 lemma funcsetI: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> f \<in> A \<rightarrow> B"
```
```    45   by (simp add: Pi_def)
```
```    46
```
```    47 lemma Pi_mem: "f \<in> Pi A B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B x"
```
```    48   by (simp add: Pi_def)
```
```    49
```
```    50 lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
```
```    51   unfolding Pi_def by auto
```
```    52
```
```    53 lemma PiE [elim]: "f \<in> Pi A B \<Longrightarrow> (f x \<in> B x \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> Q) \<Longrightarrow> Q"
```
```    54   by (auto simp: Pi_def)
```
```    55
```
```    56 lemma Pi_cong: "(\<And>w. w \<in> A \<Longrightarrow> f w = g w) \<Longrightarrow> f \<in> Pi A B \<longleftrightarrow> g \<in> Pi A B"
```
```    57   by (auto simp: Pi_def)
```
```    58
```
```    59 lemma funcset_id [simp]: "(\<lambda>x. x) \<in> A \<rightarrow> A"
```
```    60   by auto
```
```    61
```
```    62 lemma funcset_mem: "f \<in> A \<rightarrow> B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B"
```
```    63   by (simp add: Pi_def)
```
```    64
```
```    65 lemma funcset_image: "f \<in> A \<rightarrow> B \<Longrightarrow> f ` A \<subseteq> B"
```
```    66   by auto
```
```    67
```
```    68 lemma image_subset_iff_funcset: "F ` A \<subseteq> B \<longleftrightarrow> F \<in> A \<rightarrow> B"
```
```    69   by auto
```
```    70
```
```    71 lemma Pi_eq_empty[simp]: "(\<Pi> x \<in> A. B x) = {} \<longleftrightarrow> (\<exists>x\<in>A. B x = {})"
```
```    72   apply (simp add: Pi_def)
```
```    73   apply auto
```
```    74   txt \<open>Converse direction requires Axiom of Choice to exhibit a function
```
```    75   picking an element from each non-empty \<^term>\<open>B x\<close>\<close>
```
```    76   apply (drule_tac x = "\<lambda>u. SOME y. y \<in> B u" in spec)
```
```    77   apply auto
```
```    78   apply (cut_tac P = "\<lambda>y. y \<in> B x" in some_eq_ex)
```
```    79   apply auto
```
```    80   done
```
```    81
```
```    82 lemma Pi_empty [simp]: "Pi {} B = UNIV"
```
```    83   by (simp add: Pi_def)
```
```    84
```
```    85 lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)"
```
```    86   by auto
```
```    87
```
```    88 lemma Pi_UN:
```
```    89   fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
```
```    90   assumes "finite I"
```
```    91     and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
```
```    92   shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)"
```
```    93 proof (intro set_eqI iffI)
```
```    94   fix f
```
```    95   assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)"
```
```    96   then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i"
```
```    97     by auto
```
```    98   from bchoice[OF this] obtain n where n: "f i \<in> A (n i) i" if "i \<in> I" for i
```
```    99     by auto
```
```   100   obtain k where k: "n i \<le> k" if "i \<in> I" for i
```
```   101     using \<open>finite I\<close> finite_nat_set_iff_bounded_le[of "n`I"] by auto
```
```   102   have "f \<in> Pi I (A k)"
```
```   103   proof (intro Pi_I)
```
```   104     fix i
```
```   105     assume "i \<in> I"
```
```   106     from mono[OF this, of "n i" k] k[OF this] n[OF this]
```
```   107     show "f i \<in> A k i" by auto
```
```   108   qed
```
```   109   then show "f \<in> (\<Union>n. Pi I (A n))"
```
```   110     by auto
```
```   111 qed auto
```
```   112
```
```   113 lemma Pi_UNIV [simp]: "A \<rightarrow> UNIV = UNIV"
```
```   114   by (simp add: Pi_def)
```
```   115
```
```   116 text \<open>Covariance of Pi-sets in their second argument\<close>
```
```   117 lemma Pi_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> Pi A B \<subseteq> Pi A C"
```
```   118   by auto
```
```   119
```
```   120 text \<open>Contravariance of Pi-sets in their first argument\<close>
```
```   121 lemma Pi_anti_mono: "A' \<subseteq> A \<Longrightarrow> Pi A B \<subseteq> Pi A' B"
```
```   122   by auto
```
```   123
```
```   124 lemma prod_final:
```
```   125   assumes 1: "fst \<circ> f \<in> Pi A B"
```
```   126     and 2: "snd \<circ> f \<in> Pi A C"
```
```   127   shows "f \<in> (\<Pi> z \<in> A. B z \<times> C z)"
```
```   128 proof (rule Pi_I)
```
```   129   fix z
```
```   130   assume z: "z \<in> A"
```
```   131   have "f z = (fst (f z), snd (f z))"
```
```   132     by simp
```
```   133   also have "\<dots> \<in> B z \<times> C z"
```
```   134     by (metis SigmaI PiE o_apply 1 2 z)
```
```   135   finally show "f z \<in> B z \<times> C z" .
```
```   136 qed
```
```   137
```
```   138 lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X"
```
```   139   by (auto simp: Pi_def)
```
```   140
```
```   141 lemma Pi_split_insert_domain[simp]: "x \<in> Pi (insert i I) X \<longleftrightarrow> x \<in> Pi I X \<and> x i \<in> X i"
```
```   142   by (auto simp: Pi_def)
```
```   143
```
```   144 lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B"
```
```   145   by (auto simp: Pi_def)
```
```   146
```
```   147 lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
```
```   148   by (auto simp: Pi_def)
```
```   149
```
```   150 lemma Pi_fupd_iff: "i \<in> I \<Longrightarrow> f \<in> Pi I (B(i := A)) \<longleftrightarrow> f \<in> Pi (I - {i}) B \<and> f i \<in> A"
```
```   151   apply auto
```
```   152   apply (drule_tac x=x in Pi_mem)
```
```   153   apply (simp_all split: if_split_asm)
```
```   154   apply (drule_tac x=i in Pi_mem)
```
```   155   apply (auto dest!: Pi_mem)
```
```   156   done
```
```   157
```
```   158
```
```   159 subsection \<open>Composition With a Restricted Domain: \<^term>\<open>compose\<close>\<close>
```
```   160
```
```   161 lemma funcset_compose: "f \<in> A \<rightarrow> B \<Longrightarrow> g \<in> B \<rightarrow> C \<Longrightarrow> compose A g f \<in> A \<rightarrow> C"
```
```   162   by (simp add: Pi_def compose_def restrict_def)
```
```   163
```
```   164 lemma compose_assoc:
```
```   165   assumes "f \<in> A \<rightarrow> B"
```
```   166   shows "compose A h (compose A g f) = compose A (compose B h g) f"
```
```   167   using assms by (simp add: fun_eq_iff Pi_def compose_def restrict_def)
```
```   168
```
```   169 lemma compose_eq: "x \<in> A \<Longrightarrow> compose A g f x = g (f x)"
```
```   170   by (simp add: compose_def restrict_def)
```
```   171
```
```   172 lemma surj_compose: "f ` A = B \<Longrightarrow> g ` B = C \<Longrightarrow> compose A g f ` A = C"
```
```   173   by (auto simp add: image_def compose_eq)
```
```   174
```
```   175
```
```   176 subsection \<open>Bounded Abstraction: \<^term>\<open>restrict\<close>\<close>
```
```   177
```
```   178 lemma restrict_cong: "I = J \<Longrightarrow> (\<And>i. i \<in> J =simp=> f i = g i) \<Longrightarrow> restrict f I = restrict g J"
```
```   179   by (auto simp: restrict_def fun_eq_iff simp_implies_def)
```
```   180
```
```   181 lemma restrict_in_funcset: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> (\<lambda>x\<in>A. f x) \<in> A \<rightarrow> B"
```
```   182   by (simp add: Pi_def restrict_def)
```
```   183
```
```   184 lemma restrictI[intro!]: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> (\<lambda>x\<in>A. f x) \<in> Pi A B"
```
```   185   by (simp add: Pi_def restrict_def)
```
```   186
```
```   187 lemma restrict_apply[simp]: "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else undefined)"
```
```   188   by (simp add: restrict_def)
```
```   189
```
```   190 lemma restrict_apply': "x \<in> A \<Longrightarrow> (\<lambda>y\<in>A. f y) x = f x"
```
```   191   by simp
```
```   192
```
```   193 lemma restrict_ext: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
```
```   194   by (simp add: fun_eq_iff Pi_def restrict_def)
```
```   195
```
```   196 lemma restrict_UNIV: "restrict f UNIV = f"
```
```   197   by (simp add: restrict_def)
```
```   198
```
```   199 lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
```
```   200   by (simp add: inj_on_def restrict_def)
```
```   201
```
```   202 lemma Id_compose: "f \<in> A \<rightarrow> B \<Longrightarrow> f \<in> extensional A \<Longrightarrow> compose A (\<lambda>y\<in>B. y) f = f"
```
```   203   by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
```
```   204
```
```   205 lemma compose_Id: "g \<in> A \<rightarrow> B \<Longrightarrow> g \<in> extensional A \<Longrightarrow> compose A g (\<lambda>x\<in>A. x) = g"
```
```   206   by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
```
```   207
```
```   208 lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
```
```   209   by (auto simp add: restrict_def)
```
```   210
```
```   211 lemma restrict_restrict[simp]: "restrict (restrict f A) B = restrict f (A \<inter> B)"
```
```   212   unfolding restrict_def by (simp add: fun_eq_iff)
```
```   213
```
```   214 lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I"
```
```   215   by (auto simp: restrict_def)
```
```   216
```
```   217 lemma restrict_upd[simp]: "i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"
```
```   218   by (auto simp: fun_eq_iff)
```
```   219
```
```   220 lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A"
```
```   221   by (auto simp: restrict_def Pi_def)
```
```   222
```
```   223
```
```   224 subsection \<open>Bijections Between Sets\<close>
```
```   225
```
```   226 text \<open>The definition of \<^const>\<open>bij_betw\<close> is in \<open>Fun.thy\<close>, but most of
```
```   227 the theorems belong here, or need at least \<^term>\<open>Hilbert_Choice\<close>.\<close>
```
```   228
```
```   229 lemma bij_betwI:
```
```   230   assumes "f \<in> A \<rightarrow> B"
```
```   231     and "g \<in> B \<rightarrow> A"
```
```   232     and g_f: "\<And>x. x\<in>A \<Longrightarrow> g (f x) = x"
```
```   233     and f_g: "\<And>y. y\<in>B \<Longrightarrow> f (g y) = y"
```
```   234   shows "bij_betw f A B"
```
```   235   unfolding bij_betw_def
```
```   236 proof
```
```   237   show "inj_on f A"
```
```   238     by (metis g_f inj_on_def)
```
```   239   have "f ` A \<subseteq> B"
```
```   240     using \<open>f \<in> A \<rightarrow> B\<close> by auto
```
```   241   moreover
```
```   242   have "B \<subseteq> f ` A"
```
```   243     by auto (metis Pi_mem \<open>g \<in> B \<rightarrow> A\<close> f_g image_iff)
```
```   244   ultimately show "f ` A = B"
```
```   245     by blast
```
```   246 qed
```
```   247
```
```   248 lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
```
```   249   by (auto simp add: bij_betw_def)
```
```   250
```
```   251 lemma inj_on_compose: "bij_betw f A B \<Longrightarrow> inj_on g B \<Longrightarrow> inj_on (compose A g f) A"
```
```   252   by (auto simp add: bij_betw_def inj_on_def compose_eq)
```
```   253
```
```   254 lemma bij_betw_compose: "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (compose A g f) A C"
```
```   255   apply (simp add: bij_betw_def compose_eq inj_on_compose)
```
```   256   apply (auto simp add: compose_def image_def)
```
```   257   done
```
```   258
```
```   259 lemma bij_betw_restrict_eq [simp]: "bij_betw (restrict f A) A B = bij_betw f A B"
```
```   260   by (simp add: bij_betw_def)
```
```   261
```
```   262
```
```   263 subsection \<open>Extensionality\<close>
```
```   264
```
```   265 lemma extensional_empty[simp]: "extensional {} = {\<lambda>x. undefined}"
```
```   266   unfolding extensional_def by auto
```
```   267
```
```   268 lemma extensional_arb: "f \<in> extensional A \<Longrightarrow> x \<notin> A \<Longrightarrow> f x = undefined"
```
```   269   by (simp add: extensional_def)
```
```   270
```
```   271 lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
```
```   272   by (simp add: restrict_def extensional_def)
```
```   273
```
```   274 lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
```
```   275   by (simp add: compose_def)
```
```   276
```
```   277 lemma extensionalityI:
```
```   278   assumes "f \<in> extensional A"
```
```   279     and "g \<in> extensional A"
```
```   280     and "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
```
```   281   shows "f = g"
```
```   282   using assms by (force simp add: fun_eq_iff extensional_def)
```
```   283
```
```   284 lemma extensional_restrict:  "f \<in> extensional A \<Longrightarrow> restrict f A = f"
```
```   285   by (rule extensionalityI[OF restrict_extensional]) auto
```
```   286
```
```   287 lemma extensional_subset: "f \<in> extensional A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f \<in> extensional B"
```
```   288   unfolding extensional_def by auto
```
```   289
```
```   290 lemma inv_into_funcset: "f ` A = B \<Longrightarrow> (\<lambda>x\<in>B. inv_into A f x) \<in> B \<rightarrow> A"
```
```   291   by (unfold inv_into_def) (fast intro: someI2)
```
```   292
```
```   293 lemma compose_inv_into_id: "bij_betw f A B \<Longrightarrow> compose A (\<lambda>y\<in>B. inv_into A f y) f = (\<lambda>x\<in>A. x)"
```
```   294   apply (simp add: bij_betw_def compose_def)
```
```   295   apply (rule restrict_ext, auto)
```
```   296   done
```
```   297
```
```   298 lemma compose_id_inv_into: "f ` A = B \<Longrightarrow> compose B f (\<lambda>y\<in>B. inv_into A f y) = (\<lambda>x\<in>B. x)"
```
```   299   apply (simp add: compose_def)
```
```   300   apply (rule restrict_ext)
```
```   301   apply (simp add: f_inv_into_f)
```
```   302   done
```
```   303
```
```   304 lemma extensional_insert[intro, simp]:
```
```   305   assumes "a \<in> extensional (insert i I)"
```
```   306   shows "a(i := b) \<in> extensional (insert i I)"
```
```   307   using assms unfolding extensional_def by auto
```
```   308
```
```   309 lemma extensional_Int[simp]: "extensional I \<inter> extensional I' = extensional (I \<inter> I')"
```
```   310   unfolding extensional_def by auto
```
```   311
```
```   312 lemma extensional_UNIV[simp]: "extensional UNIV = UNIV"
```
```   313   by (auto simp: extensional_def)
```
```   314
```
```   315 lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B"
```
```   316   unfolding restrict_def extensional_def by auto
```
```   317
```
```   318 lemma extensional_insert_undefined[intro, simp]:
```
```   319   "a \<in> extensional (insert i I) \<Longrightarrow> a(i := undefined) \<in> extensional I"
```
```   320   unfolding extensional_def by auto
```
```   321
```
```   322 lemma extensional_insert_cancel[intro, simp]:
```
```   323   "a \<in> extensional I \<Longrightarrow> a \<in> extensional (insert i I)"
```
```   324   unfolding extensional_def by auto
```
```   325
```
```   326
```
```   327 subsection \<open>Cardinality\<close>
```
```   328
```
```   329 lemma card_inj: "f \<in> A \<rightarrow> B \<Longrightarrow> inj_on f A \<Longrightarrow> finite B \<Longrightarrow> card A \<le> card B"
```
```   330   by (rule card_inj_on_le) auto
```
```   331
```
```   332 lemma card_bij:
```
```   333   assumes "f \<in> A \<rightarrow> B" "inj_on f A"
```
```   334     and "g \<in> B \<rightarrow> A" "inj_on g B"
```
```   335     and "finite A" "finite B"
```
```   336   shows "card A = card B"
```
```   337   using assms by (blast intro: card_inj order_antisym)
```
```   338
```
```   339
```
```   340 subsection \<open>Extensional Function Spaces\<close>
```
```   341
```
```   342 definition PiE :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set"
```
```   343   where "PiE S T = Pi S T \<inter> extensional S"
```
```   344
```
```   345 abbreviation "Pi\<^sub>E A B \<equiv> PiE A B"
```
```   346
```
```   347 syntax
```
```   348   "_PiE" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3\<Pi>\<^sub>E _\<in>_./ _)" 10)
```
```   349 translations
```
```   350   "\<Pi>\<^sub>E x\<in>A. B" \<rightleftharpoons> "CONST Pi\<^sub>E A (\<lambda>x. B)"
```
```   351
```
```   352 abbreviation extensional_funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr "\<rightarrow>\<^sub>E" 60)
```
```   353   where "A \<rightarrow>\<^sub>E B \<equiv> (\<Pi>\<^sub>E i\<in>A. B)"
```
```   354
```
```   355 lemma extensional_funcset_def: "extensional_funcset S T = (S \<rightarrow> T) \<inter> extensional S"
```
```   356   by (simp add: PiE_def)
```
```   357
```
```   358 lemma PiE_empty_domain[simp]: "Pi\<^sub>E {} T = {\<lambda>x. undefined}"
```
```   359   unfolding PiE_def by simp
```
```   360
```
```   361 lemma PiE_UNIV_domain: "Pi\<^sub>E UNIV T = Pi UNIV T"
```
```   362   unfolding PiE_def by simp
```
```   363
```
```   364 lemma PiE_empty_range[simp]: "i \<in> I \<Longrightarrow> F i = {} \<Longrightarrow> (\<Pi>\<^sub>E i\<in>I. F i) = {}"
```
```   365   unfolding PiE_def by auto
```
```   366
```
```   367 lemma PiE_eq_empty_iff: "Pi\<^sub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})"
```
```   368 proof
```
```   369   assume "Pi\<^sub>E I F = {}"
```
```   370   show "\<exists>i\<in>I. F i = {}"
```
```   371   proof (rule ccontr)
```
```   372     assume "\<not> ?thesis"
```
```   373     then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)"
```
```   374       by auto
```
```   375     from choice[OF this]
```
```   376     obtain f where " \<forall>x. (x \<in> I \<longrightarrow> f x \<in> F x) \<and> (x \<notin> I \<longrightarrow> f x = undefined)" ..
```
```   377     then have "f \<in> Pi\<^sub>E I F"
```
```   378       by (auto simp: extensional_def PiE_def)
```
```   379     with \<open>Pi\<^sub>E I F = {}\<close> show False
```
```   380       by auto
```
```   381   qed
```
```   382 qed (auto simp: PiE_def)
```
```   383
```
```   384 lemma PiE_arb: "f \<in> Pi\<^sub>E S T \<Longrightarrow> x \<notin> S \<Longrightarrow> f x = undefined"
```
```   385   unfolding PiE_def by auto (auto dest!: extensional_arb)
```
```   386
```
```   387 lemma PiE_mem: "f \<in> Pi\<^sub>E S T \<Longrightarrow> x \<in> S \<Longrightarrow> f x \<in> T x"
```
```   388   unfolding PiE_def by auto
```
```   389
```
```   390 lemma PiE_fun_upd: "y \<in> T x \<Longrightarrow> f \<in> Pi\<^sub>E S T \<Longrightarrow> f(x := y) \<in> Pi\<^sub>E (insert x S) T"
```
```   391   unfolding PiE_def extensional_def by auto
```
```   392
```
```   393 lemma fun_upd_in_PiE: "x \<notin> S \<Longrightarrow> f \<in> Pi\<^sub>E (insert x S) T \<Longrightarrow> f(x := undefined) \<in> Pi\<^sub>E S T"
```
```   394   unfolding PiE_def extensional_def by auto
```
```   395
```
```   396 lemma PiE_insert_eq: "Pi\<^sub>E (insert x S) T = (\<lambda>(y, g). g(x := y)) ` (T x \<times> Pi\<^sub>E S T)"
```
```   397 proof -
```
```   398   {
```
```   399     fix f assume "f \<in> Pi\<^sub>E (insert x S) T" "x \<notin> S"
```
```   400     then have "f \<in> (\<lambda>(y, g). g(x := y)) ` (T x \<times> Pi\<^sub>E S T)"
```
```   401       by (auto intro!: image_eqI[where x="(f x, f(x := undefined))"] intro: fun_upd_in_PiE PiE_mem)
```
```   402   }
```
```   403   moreover
```
```   404   {
```
```   405     fix f assume "f \<in> Pi\<^sub>E (insert x S) T" "x \<in> S"
```
```   406     then have "f \<in> (\<lambda>(y, g). g(x := y)) ` (T x \<times> Pi\<^sub>E S T)"
```
```   407       by (auto intro!: image_eqI[where x="(f x, f)"] intro: fun_upd_in_PiE PiE_mem simp: insert_absorb)
```
```   408   }
```
```   409   ultimately show ?thesis
```
```   410     by (auto intro: PiE_fun_upd)
```
```   411 qed
```
```   412
```
```   413 lemma PiE_Int: "Pi\<^sub>E I A \<inter> Pi\<^sub>E I B = Pi\<^sub>E I (\<lambda>x. A x \<inter> B x)"
```
```   414   by (auto simp: PiE_def)
```
```   415
```
```   416 lemma PiE_cong: "(\<And>i. i\<in>I \<Longrightarrow> A i = B i) \<Longrightarrow> Pi\<^sub>E I A = Pi\<^sub>E I B"
```
```   417   unfolding PiE_def by (auto simp: Pi_cong)
```
```   418
```
```   419 lemma PiE_E [elim]:
```
```   420   assumes "f \<in> Pi\<^sub>E A B"
```
```   421   obtains "x \<in> A" and "f x \<in> B x"
```
```   422     | "x \<notin> A" and "f x = undefined"
```
```   423   using assms by (auto simp: Pi_def PiE_def extensional_def)
```
```   424
```
```   425 lemma PiE_I[intro!]:
```
```   426   "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> (\<And>x. x \<notin> A \<Longrightarrow> f x = undefined) \<Longrightarrow> f \<in> Pi\<^sub>E A B"
```
```   427   by (simp add: PiE_def extensional_def)
```
```   428
```
```   429 lemma PiE_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> Pi\<^sub>E A B \<subseteq> Pi\<^sub>E A C"
```
```   430   by auto
```
```   431
```
```   432 lemma PiE_iff: "f \<in> Pi\<^sub>E I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i) \<and> f \<in> extensional I"
```
```   433   by (simp add: PiE_def Pi_iff)
```
```   434
```
```   435 lemma PiE_restrict[simp]:  "f \<in> Pi\<^sub>E A B \<Longrightarrow> restrict f A = f"
```
```   436   by (simp add: extensional_restrict PiE_def)
```
```   437
```
```   438 lemma restrict_PiE[simp]: "restrict f I \<in> Pi\<^sub>E I S \<longleftrightarrow> f \<in> Pi I S"
```
```   439   by (auto simp: PiE_iff)
```
```   440
```
```   441 lemma PiE_eq_subset:
```
```   442   assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
```
```   443     and eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
```
```   444     and "i \<in> I"
```
```   445   shows "F i \<subseteq> F' i"
```
```   446 proof
```
```   447   fix x
```
```   448   assume "x \<in> F i"
```
```   449   with ne have "\<forall>j. \<exists>y. (j \<in> I \<longrightarrow> y \<in> F j \<and> (i = j \<longrightarrow> x = y)) \<and> (j \<notin> I \<longrightarrow> y = undefined)"
```
```   450     by auto
```
```   451   from choice[OF this] obtain f
```
```   452     where f: " \<forall>j. (j \<in> I \<longrightarrow> f j \<in> F j \<and> (i = j \<longrightarrow> x = f j)) \<and> (j \<notin> I \<longrightarrow> f j = undefined)" ..
```
```   453   then have "f \<in> Pi\<^sub>E I F"
```
```   454     by (auto simp: extensional_def PiE_def)
```
```   455   then have "f \<in> Pi\<^sub>E I F'"
```
```   456     using assms by simp
```
```   457   then show "x \<in> F' i"
```
```   458     using f \<open>i \<in> I\<close> by (auto simp: PiE_def)
```
```   459 qed
```
```   460
```
```   461 lemma PiE_eq_iff_not_empty:
```
```   462   assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
```
```   463   shows "Pi\<^sub>E I F = Pi\<^sub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)"
```
```   464 proof (intro iffI ballI)
```
```   465   fix i
```
```   466   assume eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
```
```   467   assume i: "i \<in> I"
```
```   468   show "F i = F' i"
```
```   469     using PiE_eq_subset[of I F F', OF ne eq i]
```
```   470     using PiE_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]
```
```   471     by auto
```
```   472 qed (auto simp: PiE_def)
```
```   473
```
```   474 lemma PiE_eq_iff:
```
```   475   "Pi\<^sub>E I F = Pi\<^sub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i) \<or> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
```
```   476 proof (intro iffI disjCI)
```
```   477   assume eq[simp]: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
```
```   478   assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
```
```   479   then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})"
```
```   480     using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by auto
```
```   481   with PiE_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i"
```
```   482     by auto
```
```   483 next
```
```   484   assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})"
```
```   485   then show "Pi\<^sub>E I F = Pi\<^sub>E I F'"
```
```   486     using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by (auto simp: PiE_def)
```
```   487 qed
```
```   488
```
```   489 lemma extensional_funcset_fun_upd_restricts_rangeI:
```
```   490   "\<forall>y \<in> S. f x \<noteq> f y \<Longrightarrow> f \<in> (insert x S) \<rightarrow>\<^sub>E T \<Longrightarrow> f(x := undefined) \<in> S \<rightarrow>\<^sub>E (T - {f x})"
```
```   491   unfolding extensional_funcset_def extensional_def
```
```   492   apply auto
```
```   493   apply (case_tac "x = xa")
```
```   494   apply auto
```
```   495   done
```
```   496
```
```   497 lemma extensional_funcset_fun_upd_extends_rangeI:
```
```   498   assumes "a \<in> T" "f \<in> S \<rightarrow>\<^sub>E (T - {a})"
```
```   499   shows "f(x := a) \<in> insert x S \<rightarrow>\<^sub>E  T"
```
```   500   using assms unfolding extensional_funcset_def extensional_def by auto
```
```   501
```
```   502 lemma subset_PiE:
```
```   503    "PiE I S \<subseteq> PiE I T \<longleftrightarrow> PiE I S = {} \<or> (\<forall>i \<in> I. S i \<subseteq> T i)" (is "?lhs \<longleftrightarrow> _ \<or> ?rhs")
```
```   504 proof (cases "PiE I S = {}")
```
```   505   case False
```
```   506   moreover have "?lhs = ?rhs"
```
```   507   proof
```
```   508     assume L: ?lhs
```
```   509     have "\<And>i. i\<in>I \<Longrightarrow> S i \<noteq> {}"
```
```   510       using False PiE_eq_empty_iff by blast
```
```   511     with L show ?rhs
```
```   512       by (simp add: PiE_Int PiE_eq_iff inf.absorb_iff2)
```
```   513   qed auto
```
```   514   ultimately show ?thesis
```
```   515     by simp
```
```   516 qed simp
```
```   517
```
```   518 lemma PiE_eq:
```
```   519    "PiE I S = PiE I T \<longleftrightarrow> PiE I S = {} \<and> PiE I T = {} \<or> (\<forall>i \<in> I. S i = T i)"
```
```   520   by (auto simp: PiE_eq_iff PiE_eq_empty_iff)
```
```   521
```
```   522 lemma PiE_UNIV [simp]: "PiE UNIV (\<lambda>i. UNIV) = UNIV"
```
```   523   by blast
```
```   524
```
```   525 lemma image_projection_PiE:
```
```   526   "(\<lambda>f. f i) ` (PiE I S) = (if PiE I S = {} then {} else if i \<in> I then S i else {undefined})"
```
```   527 proof -
```
```   528   have "(\<lambda>f. f i) ` Pi\<^sub>E I S = S i" if "i \<in> I" "f \<in> PiE I S" for f
```
```   529     using that apply auto
```
```   530     by (rule_tac x="(\<lambda>k. if k=i then x else f k)" in image_eqI) auto
```
```   531   moreover have "(\<lambda>f. f i) ` Pi\<^sub>E I S = {undefined}" if "f \<in> PiE I S" "i \<notin> I" for f
```
```   532     using that by (blast intro: PiE_arb [OF that, symmetric])
```
```   533   ultimately show ?thesis
```
```   534     by auto
```
```   535 qed
```
```   536
```
```   537 lemma PiE_singleton:
```
```   538   assumes "f \<in> extensional A"
```
```   539   shows   "PiE A (\<lambda>x. {f x}) = {f}"
```
```   540 proof -
```
```   541   {
```
```   542     fix g assume "g \<in> PiE A (\<lambda>x. {f x})"
```
```   543     hence "g x = f x" for x
```
```   544       using assms by (cases "x \<in> A") (auto simp: extensional_def)
```
```   545     hence "g = f" by (simp add: fun_eq_iff)
```
```   546   }
```
```   547   thus ?thesis using assms by (auto simp: extensional_def)
```
```   548 qed
```
```   549
```
```   550 lemma PiE_eq_singleton: "(\<Pi>\<^sub>E i\<in>I. S i) = {\<lambda>i\<in>I. f i} \<longleftrightarrow> (\<forall>i\<in>I. S i = {f i})"
```
```   551   by (metis (mono_tags, lifting) PiE_eq PiE_singleton insert_not_empty restrict_apply' restrict_extensional)
```
```   552
```
```   553 lemma PiE_over_singleton_iff: "(\<Pi>\<^sub>E x\<in>{a}. B x) = (\<Union>b \<in> B a. {\<lambda>x \<in> {a}. b})"
```
```   554   apply (auto simp: PiE_iff split: if_split_asm)
```
```   555   apply (metis (no_types, lifting) extensionalityI restrict_apply' restrict_extensional singletonD)
```
```   556   done
```
```   557
```
```   558 lemma all_PiE_elements:
```
```   559    "(\<forall>z \<in> PiE I S. \<forall>i \<in> I. P i (z i)) \<longleftrightarrow> PiE I S = {} \<or> (\<forall>i \<in> I. \<forall>x \<in> S i. P i x)" (is "?lhs = ?rhs")
```
```   560 proof (cases "PiE I S = {}")
```
```   561   case False
```
```   562   then obtain f where f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> S i"
```
```   563     by fastforce
```
```   564   show ?thesis
```
```   565   proof
```
```   566     assume L: ?lhs
```
```   567     have "P i x"
```
```   568       if "i \<in> I" "x \<in> S i" for i x
```
```   569     proof -
```
```   570       have "(\<lambda>j \<in> I. if j=i then x else f j) \<in> PiE I S"
```
```   571         by (simp add: f that(2))
```
```   572       then have "P i ((\<lambda>j \<in> I. if j=i then x else f j) i)"
```
```   573         using L that(1) by blast
```
```   574       with that show ?thesis
```
```   575         by simp
```
```   576     qed
```
```   577     then show ?rhs
```
```   578       by (simp add: False)
```
```   579   qed fastforce
```
```   580 qed simp
```
```   581
```
```   582 lemma PiE_ext: "\<lbrakk>x \<in> PiE k s; y \<in> PiE k s; \<And>i. i \<in> k \<Longrightarrow> x i = y i\<rbrakk> \<Longrightarrow> x = y"
```
```   583   by (metis ext PiE_E)
```
```   584
```
```   585
```
```   586 subsubsection \<open>Injective Extensional Function Spaces\<close>
```
```   587
```
```   588 lemma extensional_funcset_fun_upd_inj_onI:
```
```   589   assumes "f \<in> S \<rightarrow>\<^sub>E (T - {a})"
```
```   590     and "inj_on f S"
```
```   591   shows "inj_on (f(x := a)) S"
```
```   592   using assms
```
```   593   unfolding extensional_funcset_def by (auto intro!: inj_on_fun_updI)
```
```   594
```
```   595 lemma extensional_funcset_extend_domain_inj_on_eq:
```
```   596   assumes "x \<notin> S"
```
```   597   shows "{f. f \<in> (insert x S) \<rightarrow>\<^sub>E T \<and> inj_on f (insert x S)} =
```
```   598     (\<lambda>(y, g). g(x:=y)) ` {(y, g). y \<in> T \<and> g \<in> S \<rightarrow>\<^sub>E (T - {y}) \<and> inj_on g S}"
```
```   599   using assms
```
```   600   apply (auto del: PiE_I PiE_E)
```
```   601   apply (auto intro: extensional_funcset_fun_upd_inj_onI
```
```   602     extensional_funcset_fun_upd_extends_rangeI del: PiE_I PiE_E)
```
```   603   apply (auto simp add: image_iff inj_on_def)
```
```   604   apply (rule_tac x="xa x" in exI)
```
```   605   apply (auto intro: PiE_mem del: PiE_I PiE_E)
```
```   606   apply (rule_tac x="xa(x := undefined)" in exI)
```
```   607   apply (auto intro!: extensional_funcset_fun_upd_restricts_rangeI)
```
```   608   apply (auto dest!: PiE_mem split: if_split_asm)
```
```   609   done
```
```   610
```
```   611 lemma extensional_funcset_extend_domain_inj_onI:
```
```   612   assumes "x \<notin> S"
```
```   613   shows "inj_on (\<lambda>(y, g). g(x := y)) {(y, g). y \<in> T \<and> g \<in> S \<rightarrow>\<^sub>E (T - {y}) \<and> inj_on g S}"
```
```   614   using assms
```
```   615   apply (auto intro!: inj_onI)
```
```   616   apply (metis fun_upd_same)
```
```   617   apply (metis assms PiE_arb fun_upd_triv fun_upd_upd)
```
```   618   done
```
```   619
```
```   620
```
```   621 subsubsection \<open>Misc properties of functions, composition and restriction from HOL Light\<close>
```
```   622
```
```   623 lemma function_factors_left_gen:
```
```   624   "(\<forall>x y. P x \<and> P y \<and> g x = g y \<longrightarrow> f x = f y) \<longleftrightarrow> (\<exists>h. \<forall>x. P x \<longrightarrow> f x = h(g x))"
```
```   625   (is "?lhs = ?rhs")
```
```   626 proof
```
```   627   assume L: ?lhs
```
```   628   then show ?rhs
```
```   629     apply (rule_tac x="f \<circ> inv_into (Collect P) g" in exI)
```
```   630     unfolding o_def
```
```   631     by (metis (mono_tags, hide_lams) f_inv_into_f imageI inv_into_into mem_Collect_eq)
```
```   632 qed auto
```
```   633
```
```   634 lemma function_factors_left:
```
```   635   "(\<forall>x y. (g x = g y) \<longrightarrow> (f x = f y)) \<longleftrightarrow> (\<exists>h. f = h \<circ> g)"
```
```   636   using function_factors_left_gen [of "\<lambda>x. True" g f] unfolding o_def by blast
```
```   637
```
```   638 lemma function_factors_right_gen:
```
```   639   "(\<forall>x. P x \<longrightarrow> (\<exists>y. g y = f x)) \<longleftrightarrow> (\<exists>h. \<forall>x. P x \<longrightarrow> f x = g(h x))"
```
```   640   by metis
```
```   641
```
```   642 lemma function_factors_right:
```
```   643   "(\<forall>x. \<exists>y. g y = f x) \<longleftrightarrow> (\<exists>h. f = g \<circ> h)"
```
```   644   unfolding o_def by metis
```
```   645
```
```   646 lemma restrict_compose_right:
```
```   647    "restrict (g \<circ> restrict f S) S = restrict (g \<circ> f) S"
```
```   648   by auto
```
```   649
```
```   650 lemma restrict_compose_left:
```
```   651    "f ` S \<subseteq> T \<Longrightarrow> restrict (restrict g T \<circ> f) S = restrict (g \<circ> f) S"
```
```   652   by fastforce
```
```   653
```
```   654
```
```   655 subsubsection \<open>Cardinality\<close>
```
```   656
```
```   657 lemma finite_PiE: "finite S \<Longrightarrow> (\<And>i. i \<in> S \<Longrightarrow> finite (T i)) \<Longrightarrow> finite (\<Pi>\<^sub>E i \<in> S. T i)"
```
```   658   by (induct S arbitrary: T rule: finite_induct) (simp_all add: PiE_insert_eq)
```
```   659
```
```   660 lemma inj_combinator: "x \<notin> S \<Longrightarrow> inj_on (\<lambda>(y, g). g(x := y)) (T x \<times> Pi\<^sub>E S T)"
```
```   661 proof (safe intro!: inj_onI ext)
```
```   662   fix f y g z
```
```   663   assume "x \<notin> S"
```
```   664   assume fg: "f \<in> Pi\<^sub>E S T" "g \<in> Pi\<^sub>E S T"
```
```   665   assume "f(x := y) = g(x := z)"
```
```   666   then have *: "\<And>i. (f(x := y)) i = (g(x := z)) i"
```
```   667     unfolding fun_eq_iff by auto
```
```   668   from this[of x] show "y = z" by simp
```
```   669   fix i from *[of i] \<open>x \<notin> S\<close> fg show "f i = g i"
```
```   670     by (auto split: if_split_asm simp: PiE_def extensional_def)
```
```   671 qed
```
```   672
```
```   673 lemma card_PiE: "finite S \<Longrightarrow> card (\<Pi>\<^sub>E i \<in> S. T i) = (\<Prod> i\<in>S. card (T i))"
```
```   674 proof (induct rule: finite_induct)
```
```   675   case empty
```
```   676   then show ?case by auto
```
```   677 next
```
```   678   case (insert x S)
```
```   679   then show ?case
```
```   680     by (simp add: PiE_insert_eq inj_combinator card_image card_cartesian_product)
```
```   681 qed
```
```   682
```
```   683 end
```