src/HOL/Library/FuncSet.thy
 author nipkow Tue Sep 22 14:31:22 2015 +0200 (2015-09-22) changeset 61225 1a690dce8cfc parent 59425 c5e79df8cc21 child 61359 e985b52c3eb3 permissions -rw-r--r--
tuned references
```     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 Hilbert_Choice Main
```
```     9 begin
```
```    10
```
```    11 definition Pi :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set"
```
```    12   where "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}"
```
```    13
```
```    14 definition extensional :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) set"
```
```    15   where "extensional A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = undefined}"
```
```    16
```
```    17 definition "restrict" :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
```
```    18   where "restrict f A = (\<lambda>x. if x \<in> A then f x else undefined)"
```
```    19
```
```    20 abbreviation funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  (infixr "->" 60)
```
```    21   where "A -> B \<equiv> Pi A (\<lambda>_. B)"
```
```    22
```
```    23 notation (xsymbols)
```
```    24   funcset  (infixr "\<rightarrow>" 60)
```
```    25
```
```    26 syntax
```
```    27   "_Pi"  :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3PI _:_./ _)" 10)
```
```    28   "_lam" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> 'b)"  ("(3%_:_./ _)" [0,0,3] 3)
```
```    29 syntax (xsymbols)
```
```    30   "_Pi" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
```
```    31   "_lam" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
```
```    32 syntax (HTML output)
```
```    33   "_Pi" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
```
```    34   "_lam" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
```
```    35 translations
```
```    36   "\<Pi> x\<in>A. B" \<rightleftharpoons> "CONST Pi A (\<lambda>x. B)"
```
```    37   "\<lambda>x\<in>A. f" \<rightleftharpoons> "CONST restrict (\<lambda>x. f) A"
```
```    38
```
```    39 definition "compose" :: "'a set \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'c)"
```
```    40   where "compose A g f = (\<lambda>x\<in>A. g (f x))"
```
```    41
```
```    42
```
```    43 subsection \<open>Basic Properties of @{term Pi}\<close>
```
```    44
```
```    45 lemma Pi_I[intro!]: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi A B"
```
```    46   by (simp add: Pi_def)
```
```    47
```
```    48 lemma Pi_I'[simp]: "(\<And>x. x \<in> A \<longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi A B"
```
```    49   by (simp add:Pi_def)
```
```    50
```
```    51 lemma funcsetI: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> f \<in> A \<rightarrow> B"
```
```    52   by (simp add: Pi_def)
```
```    53
```
```    54 lemma Pi_mem: "f \<in> Pi A B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B x"
```
```    55   by (simp add: Pi_def)
```
```    56
```
```    57 lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
```
```    58   unfolding Pi_def by auto
```
```    59
```
```    60 lemma PiE [elim]: "f \<in> Pi A B \<Longrightarrow> (f x \<in> B x \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> Q) \<Longrightarrow> Q"
```
```    61   by (auto simp: Pi_def)
```
```    62
```
```    63 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"
```
```    64   by (auto simp: Pi_def)
```
```    65
```
```    66 lemma funcset_id [simp]: "(\<lambda>x. x) \<in> A \<rightarrow> A"
```
```    67   by auto
```
```    68
```
```    69 lemma funcset_mem: "f \<in> A \<rightarrow> B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B"
```
```    70   by (simp add: Pi_def)
```
```    71
```
```    72 lemma funcset_image: "f \<in> A \<rightarrow> B \<Longrightarrow> f ` A \<subseteq> B"
```
```    73   by auto
```
```    74
```
```    75 lemma image_subset_iff_funcset: "F ` A \<subseteq> B \<longleftrightarrow> F \<in> A \<rightarrow> B"
```
```    76   by auto
```
```    77
```
```    78 lemma Pi_eq_empty[simp]: "(\<Pi> x \<in> A. B x) = {} \<longleftrightarrow> (\<exists>x\<in>A. B x = {})"
```
```    79   apply (simp add: Pi_def)
```
```    80   apply auto
```
```    81   txt \<open>Converse direction requires Axiom of Choice to exhibit a function
```
```    82   picking an element from each non-empty @{term "B x"}\<close>
```
```    83   apply (drule_tac x = "\<lambda>u. SOME y. y \<in> B u" in spec)
```
```    84   apply auto
```
```    85   apply (cut_tac P = "\<lambda>y. y \<in> B x" in some_eq_ex)
```
```    86   apply auto
```
```    87   done
```
```    88
```
```    89 lemma Pi_empty [simp]: "Pi {} B = UNIV"
```
```    90   by (simp add: Pi_def)
```
```    91
```
```    92 lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)"
```
```    93   by auto
```
```    94
```
```    95 lemma Pi_UN:
```
```    96   fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
```
```    97   assumes "finite I"
```
```    98     and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
```
```    99   shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)"
```
```   100 proof (intro set_eqI iffI)
```
```   101   fix f
```
```   102   assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)"
```
```   103   then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i"
```
```   104     by auto
```
```   105   from bchoice[OF this] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)"
```
```   106     by auto
```
```   107   obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k"
```
```   108     using \<open>finite I\<close> finite_nat_set_iff_bounded_le[of "n`I"] by auto
```
```   109   have "f \<in> Pi I (A k)"
```
```   110   proof (intro Pi_I)
```
```   111     fix i
```
```   112     assume "i \<in> I"
```
```   113     from mono[OF this, of "n i" k] k[OF this] n[OF this]
```
```   114     show "f i \<in> A k i" by auto
```
```   115   qed
```
```   116   then show "f \<in> (\<Union>n. Pi I (A n))"
```
```   117     by auto
```
```   118 qed auto
```
```   119
```
```   120 lemma Pi_UNIV [simp]: "A \<rightarrow> UNIV = UNIV"
```
```   121   by (simp add: Pi_def)
```
```   122
```
```   123 text \<open>Covariance of Pi-sets in their second argument\<close>
```
```   124 lemma Pi_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> Pi A B \<subseteq> Pi A C"
```
```   125   by auto
```
```   126
```
```   127 text \<open>Contravariance of Pi-sets in their first argument\<close>
```
```   128 lemma Pi_anti_mono: "A' \<subseteq> A \<Longrightarrow> Pi A B \<subseteq> Pi A' B"
```
```   129   by auto
```
```   130
```
```   131 lemma prod_final:
```
```   132   assumes 1: "fst \<circ> f \<in> Pi A B"
```
```   133     and 2: "snd \<circ> f \<in> Pi A C"
```
```   134   shows "f \<in> (\<Pi> z \<in> A. B z \<times> C z)"
```
```   135 proof (rule Pi_I)
```
```   136   fix z
```
```   137   assume z: "z \<in> A"
```
```   138   have "f z = (fst (f z), snd (f z))"
```
```   139     by simp
```
```   140   also have "\<dots> \<in> B z \<times> C z"
```
```   141     by (metis SigmaI PiE o_apply 1 2 z)
```
```   142   finally show "f z \<in> B z \<times> C z" .
```
```   143 qed
```
```   144
```
```   145 lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X"
```
```   146   by (auto simp: Pi_def)
```
```   147
```
```   148 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"
```
```   149   by (auto simp: Pi_def)
```
```   150
```
```   151 lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B"
```
```   152   by (auto simp: Pi_def)
```
```   153
```
```   154 lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
```
```   155   by (auto simp: Pi_def)
```
```   156
```
```   157 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"
```
```   158   apply auto
```
```   159   apply (drule_tac x=x in Pi_mem)
```
```   160   apply (simp_all split: split_if_asm)
```
```   161   apply (drule_tac x=i in Pi_mem)
```
```   162   apply (auto dest!: Pi_mem)
```
```   163   done
```
```   164
```
```   165
```
```   166 subsection \<open>Composition With a Restricted Domain: @{term compose}\<close>
```
```   167
```
```   168 lemma funcset_compose: "f \<in> A \<rightarrow> B \<Longrightarrow> g \<in> B \<rightarrow> C \<Longrightarrow> compose A g f \<in> A \<rightarrow> C"
```
```   169   by (simp add: Pi_def compose_def restrict_def)
```
```   170
```
```   171 lemma compose_assoc:
```
```   172   assumes "f \<in> A \<rightarrow> B"
```
```   173     and "g \<in> B \<rightarrow> C"
```
```   174     and "h \<in> C \<rightarrow> D"
```
```   175   shows "compose A h (compose A g f) = compose A (compose B h g) f"
```
```   176   using assms by (simp add: fun_eq_iff Pi_def compose_def restrict_def)
```
```   177
```
```   178 lemma compose_eq: "x \<in> A \<Longrightarrow> compose A g f x = g (f x)"
```
```   179   by (simp add: compose_def restrict_def)
```
```   180
```
```   181 lemma surj_compose: "f ` A = B \<Longrightarrow> g ` B = C \<Longrightarrow> compose A g f ` A = C"
```
```   182   by (auto simp add: image_def compose_eq)
```
```   183
```
```   184
```
```   185 subsection \<open>Bounded Abstraction: @{term restrict}\<close>
```
```   186
```
```   187 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"
```
```   188   by (simp add: Pi_def restrict_def)
```
```   189
```
```   190 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"
```
```   191   by (simp add: Pi_def restrict_def)
```
```   192
```
```   193 lemma restrict_apply[simp]: "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else undefined)"
```
```   194   by (simp add: restrict_def)
```
```   195
```
```   196 lemma restrict_apply': "x \<in> A \<Longrightarrow> (\<lambda>y\<in>A. f y) x = f x"
```
```   197   by simp
```
```   198
```
```   199 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)"
```
```   200   by (simp add: fun_eq_iff Pi_def restrict_def)
```
```   201
```
```   202 lemma restrict_UNIV: "restrict f UNIV = f"
```
```   203   by (simp add: restrict_def)
```
```   204
```
```   205 lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
```
```   206   by (simp add: inj_on_def restrict_def)
```
```   207
```
```   208 lemma Id_compose: "f \<in> A \<rightarrow> B \<Longrightarrow> f \<in> extensional A \<Longrightarrow> compose A (\<lambda>y\<in>B. y) f = f"
```
```   209   by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
```
```   210
```
```   211 lemma compose_Id: "g \<in> A \<rightarrow> B \<Longrightarrow> g \<in> extensional A \<Longrightarrow> compose A g (\<lambda>x\<in>A. x) = g"
```
```   212   by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
```
```   213
```
```   214 lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
```
```   215   by (auto simp add: restrict_def)
```
```   216
```
```   217 lemma restrict_restrict[simp]: "restrict (restrict f A) B = restrict f (A \<inter> B)"
```
```   218   unfolding restrict_def by (simp add: fun_eq_iff)
```
```   219
```
```   220 lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I"
```
```   221   by (auto simp: restrict_def)
```
```   222
```
```   223 lemma restrict_upd[simp]: "i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"
```
```   224   by (auto simp: fun_eq_iff)
```
```   225
```
```   226 lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A"
```
```   227   by (auto simp: restrict_def Pi_def)
```
```   228
```
```   229
```
```   230 subsection \<open>Bijections Between Sets\<close>
```
```   231
```
```   232 text \<open>The definition of @{const bij_betw} is in @{text "Fun.thy"}, but most of
```
```   233 the theorems belong here, or need at least @{term Hilbert_Choice}.\<close>
```
```   234
```
```   235 lemma bij_betwI:
```
```   236   assumes "f \<in> A \<rightarrow> B"
```
```   237     and "g \<in> B \<rightarrow> A"
```
```   238     and g_f: "\<And>x. x\<in>A \<Longrightarrow> g (f x) = x"
```
```   239     and f_g: "\<And>y. y\<in>B \<Longrightarrow> f (g y) = y"
```
```   240   shows "bij_betw f A B"
```
```   241   unfolding bij_betw_def
```
```   242 proof
```
```   243   show "inj_on f A"
```
```   244     by (metis g_f inj_on_def)
```
```   245   have "f ` A \<subseteq> B"
```
```   246     using \<open>f \<in> A \<rightarrow> B\<close> by auto
```
```   247   moreover
```
```   248   have "B \<subseteq> f ` A"
```
```   249     by auto (metis Pi_mem \<open>g \<in> B \<rightarrow> A\<close> f_g image_iff)
```
```   250   ultimately show "f ` A = B"
```
```   251     by blast
```
```   252 qed
```
```   253
```
```   254 lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
```
```   255   by (auto simp add: bij_betw_def)
```
```   256
```
```   257 lemma inj_on_compose: "bij_betw f A B \<Longrightarrow> inj_on g B \<Longrightarrow> inj_on (compose A g f) A"
```
```   258   by (auto simp add: bij_betw_def inj_on_def compose_eq)
```
```   259
```
```   260 lemma bij_betw_compose: "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (compose A g f) A C"
```
```   261   apply (simp add: bij_betw_def compose_eq inj_on_compose)
```
```   262   apply (auto simp add: compose_def image_def)
```
```   263   done
```
```   264
```
```   265 lemma bij_betw_restrict_eq [simp]: "bij_betw (restrict f A) A B = bij_betw f A B"
```
```   266   by (simp add: bij_betw_def)
```
```   267
```
```   268
```
```   269 subsection \<open>Extensionality\<close>
```
```   270
```
```   271 lemma extensional_empty[simp]: "extensional {} = {\<lambda>x. undefined}"
```
```   272   unfolding extensional_def by auto
```
```   273
```
```   274 lemma extensional_arb: "f \<in> extensional A \<Longrightarrow> x \<notin> A \<Longrightarrow> f x = undefined"
```
```   275   by (simp add: extensional_def)
```
```   276
```
```   277 lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
```
```   278   by (simp add: restrict_def extensional_def)
```
```   279
```
```   280 lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
```
```   281   by (simp add: compose_def)
```
```   282
```
```   283 lemma extensionalityI:
```
```   284   assumes "f \<in> extensional A"
```
```   285     and "g \<in> extensional A"
```
```   286     and "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
```
```   287   shows "f = g"
```
```   288   using assms by (force simp add: fun_eq_iff extensional_def)
```
```   289
```
```   290 lemma extensional_restrict:  "f \<in> extensional A \<Longrightarrow> restrict f A = f"
```
```   291   by (rule extensionalityI[OF restrict_extensional]) auto
```
```   292
```
```   293 lemma extensional_subset: "f \<in> extensional A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f \<in> extensional B"
```
```   294   unfolding extensional_def by auto
```
```   295
```
```   296 lemma inv_into_funcset: "f ` A = B \<Longrightarrow> (\<lambda>x\<in>B. inv_into A f x) \<in> B \<rightarrow> A"
```
```   297   by (unfold inv_into_def) (fast intro: someI2)
```
```   298
```
```   299 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)"
```
```   300   apply (simp add: bij_betw_def compose_def)
```
```   301   apply (rule restrict_ext, auto)
```
```   302   done
```
```   303
```
```   304 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)"
```
```   305   apply (simp add: compose_def)
```
```   306   apply (rule restrict_ext)
```
```   307   apply (simp add: f_inv_into_f)
```
```   308   done
```
```   309
```
```   310 lemma extensional_insert[intro, simp]:
```
```   311   assumes "a \<in> extensional (insert i I)"
```
```   312   shows "a(i := b) \<in> extensional (insert i I)"
```
```   313   using assms unfolding extensional_def by auto
```
```   314
```
```   315 lemma extensional_Int[simp]: "extensional I \<inter> extensional I' = extensional (I \<inter> I')"
```
```   316   unfolding extensional_def by auto
```
```   317
```
```   318 lemma extensional_UNIV[simp]: "extensional UNIV = UNIV"
```
```   319   by (auto simp: extensional_def)
```
```   320
```
```   321 lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B"
```
```   322   unfolding restrict_def extensional_def by auto
```
```   323
```
```   324 lemma extensional_insert_undefined[intro, simp]:
```
```   325   "a \<in> extensional (insert i I) \<Longrightarrow> a(i := undefined) \<in> extensional I"
```
```   326   unfolding extensional_def by auto
```
```   327
```
```   328 lemma extensional_insert_cancel[intro, simp]:
```
```   329   "a \<in> extensional I \<Longrightarrow> a \<in> extensional (insert i I)"
```
```   330   unfolding extensional_def by auto
```
```   331
```
```   332
```
```   333 subsection \<open>Cardinality\<close>
```
```   334
```
```   335 lemma card_inj: "f \<in> A \<rightarrow> B \<Longrightarrow> inj_on f A \<Longrightarrow> finite B \<Longrightarrow> card A \<le> card B"
```
```   336   by (rule card_inj_on_le) auto
```
```   337
```
```   338 lemma card_bij:
```
```   339   assumes "f \<in> A \<rightarrow> B" "inj_on f A"
```
```   340     and "g \<in> B \<rightarrow> A" "inj_on g B"
```
```   341     and "finite A" "finite B"
```
```   342   shows "card A = card B"
```
```   343   using assms by (blast intro: card_inj order_antisym)
```
```   344
```
```   345
```
```   346 subsection \<open>Extensional Function Spaces\<close>
```
```   347
```
```   348 definition PiE :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set"
```
```   349   where "PiE S T = Pi S T \<inter> extensional S"
```
```   350
```
```   351 abbreviation "Pi\<^sub>E A B \<equiv> PiE A B"
```
```   352
```
```   353 syntax
```
```   354   "_PiE" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3PIE _:_./ _)" 10)
```
```   355 syntax (xsymbols)
```
```   356   "_PiE" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3\<Pi>\<^sub>E _\<in>_./ _)" 10)
```
```   357 syntax (HTML output)
```
```   358   "_PiE" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3\<Pi>\<^sub>E _\<in>_./ _)" 10)
```
```   359 translations "\<Pi>\<^sub>E x\<in>A. B" \<rightleftharpoons> "CONST Pi\<^sub>E A (\<lambda>x. B)"
```
```   360
```
```   361 abbreviation extensional_funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr "->\<^sub>E" 60)
```
```   362   where "A ->\<^sub>E B \<equiv> (\<Pi>\<^sub>E i\<in>A. B)"
```
```   363
```
```   364 notation (xsymbols)
```
```   365   extensional_funcset  (infixr "\<rightarrow>\<^sub>E" 60)
```
```   366
```
```   367 lemma extensional_funcset_def: "extensional_funcset S T = (S \<rightarrow> T) \<inter> extensional S"
```
```   368   by (simp add: PiE_def)
```
```   369
```
```   370 lemma PiE_empty_domain[simp]: "PiE {} T = {\<lambda>x. undefined}"
```
```   371   unfolding PiE_def by simp
```
```   372
```
```   373 lemma PiE_UNIV_domain: "PiE UNIV T = Pi UNIV T"
```
```   374   unfolding PiE_def by simp
```
```   375
```
```   376 lemma PiE_empty_range[simp]: "i \<in> I \<Longrightarrow> F i = {} \<Longrightarrow> (\<Pi>\<^sub>E i\<in>I. F i) = {}"
```
```   377   unfolding PiE_def by auto
```
```   378
```
```   379 lemma PiE_eq_empty_iff: "Pi\<^sub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})"
```
```   380 proof
```
```   381   assume "Pi\<^sub>E I F = {}"
```
```   382   show "\<exists>i\<in>I. F i = {}"
```
```   383   proof (rule ccontr)
```
```   384     assume "\<not> ?thesis"
```
```   385     then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)"
```
```   386       by auto
```
```   387     from choice[OF this]
```
```   388     obtain f where " \<forall>x. (x \<in> I \<longrightarrow> f x \<in> F x) \<and> (x \<notin> I \<longrightarrow> f x = undefined)" ..
```
```   389     then have "f \<in> Pi\<^sub>E I F"
```
```   390       by (auto simp: extensional_def PiE_def)
```
```   391     with \<open>Pi\<^sub>E I F = {}\<close> show False
```
```   392       by auto
```
```   393   qed
```
```   394 qed (auto simp: PiE_def)
```
```   395
```
```   396 lemma PiE_arb: "f \<in> PiE S T \<Longrightarrow> x \<notin> S \<Longrightarrow> f x = undefined"
```
```   397   unfolding PiE_def by auto (auto dest!: extensional_arb)
```
```   398
```
```   399 lemma PiE_mem: "f \<in> PiE S T \<Longrightarrow> x \<in> S \<Longrightarrow> f x \<in> T x"
```
```   400   unfolding PiE_def by auto
```
```   401
```
```   402 lemma PiE_fun_upd: "y \<in> T x \<Longrightarrow> f \<in> PiE S T \<Longrightarrow> f(x := y) \<in> PiE (insert x S) T"
```
```   403   unfolding PiE_def extensional_def by auto
```
```   404
```
```   405 lemma fun_upd_in_PiE: "x \<notin> S \<Longrightarrow> f \<in> PiE (insert x S) T \<Longrightarrow> f(x := undefined) \<in> PiE S T"
```
```   406   unfolding PiE_def extensional_def by auto
```
```   407
```
```   408 lemma PiE_insert_eq: "PiE (insert x S) T = (\<lambda>(y, g). g(x := y)) ` (T x \<times> PiE S T)"
```
```   409 proof -
```
```   410   {
```
```   411     fix f assume "f \<in> PiE (insert x S) T" "x \<notin> S"
```
```   412     with assms have "f \<in> (\<lambda>(y, g). g(x := y)) ` (T x \<times> PiE S T)"
```
```   413       by (auto intro!: image_eqI[where x="(f x, f(x := undefined))"] intro: fun_upd_in_PiE PiE_mem)
```
```   414   }
```
```   415   moreover
```
```   416   {
```
```   417     fix f assume "f \<in> PiE (insert x S) T" "x \<in> S"
```
```   418     with assms have "f \<in> (\<lambda>(y, g). g(x := y)) ` (T x \<times> PiE S T)"
```
```   419       by (auto intro!: image_eqI[where x="(f x, f)"] intro: fun_upd_in_PiE PiE_mem simp: insert_absorb)
```
```   420   }
```
```   421   ultimately show ?thesis
```
```   422     using assms by (auto intro: PiE_fun_upd)
```
```   423 qed
```
```   424
```
```   425 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)"
```
```   426   by (auto simp: PiE_def)
```
```   427
```
```   428 lemma PiE_cong: "(\<And>i. i\<in>I \<Longrightarrow> A i = B i) \<Longrightarrow> Pi\<^sub>E I A = Pi\<^sub>E I B"
```
```   429   unfolding PiE_def by (auto simp: Pi_cong)
```
```   430
```
```   431 lemma PiE_E [elim]:
```
```   432   assumes "f \<in> PiE A B"
```
```   433   obtains "x \<in> A" and "f x \<in> B x"
```
```   434     | "x \<notin> A" and "f x = undefined"
```
```   435   using assms by (auto simp: Pi_def PiE_def extensional_def)
```
```   436
```
```   437 lemma PiE_I[intro!]:
```
```   438   "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> (\<And>x. x \<notin> A \<Longrightarrow> f x = undefined) \<Longrightarrow> f \<in> PiE A B"
```
```   439   by (simp add: PiE_def extensional_def)
```
```   440
```
```   441 lemma PiE_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> PiE A B \<subseteq> PiE A C"
```
```   442   by auto
```
```   443
```
```   444 lemma PiE_iff: "f \<in> PiE I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i) \<and> f \<in> extensional I"
```
```   445   by (simp add: PiE_def Pi_iff)
```
```   446
```
```   447 lemma PiE_restrict[simp]:  "f \<in> PiE A B \<Longrightarrow> restrict f A = f"
```
```   448   by (simp add: extensional_restrict PiE_def)
```
```   449
```
```   450 lemma restrict_PiE[simp]: "restrict f I \<in> PiE I S \<longleftrightarrow> f \<in> Pi I S"
```
```   451   by (auto simp: PiE_iff)
```
```   452
```
```   453 lemma PiE_eq_subset:
```
```   454   assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
```
```   455     and eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
```
```   456     and "i \<in> I"
```
```   457   shows "F i \<subseteq> F' i"
```
```   458 proof
```
```   459   fix x
```
```   460   assume "x \<in> F i"
```
```   461   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)"
```
```   462     by auto
```
```   463   from choice[OF this] obtain f
```
```   464     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)" ..
```
```   465   then have "f \<in> Pi\<^sub>E I F"
```
```   466     by (auto simp: extensional_def PiE_def)
```
```   467   then have "f \<in> Pi\<^sub>E I F'"
```
```   468     using assms by simp
```
```   469   then show "x \<in> F' i"
```
```   470     using f \<open>i \<in> I\<close> by (auto simp: PiE_def)
```
```   471 qed
```
```   472
```
```   473 lemma PiE_eq_iff_not_empty:
```
```   474   assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
```
```   475   shows "Pi\<^sub>E I F = Pi\<^sub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)"
```
```   476 proof (intro iffI ballI)
```
```   477   fix i
```
```   478   assume eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
```
```   479   assume i: "i \<in> I"
```
```   480   show "F i = F' i"
```
```   481     using PiE_eq_subset[of I F F', OF ne eq i]
```
```   482     using PiE_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]
```
```   483     by auto
```
```   484 qed (auto simp: PiE_def)
```
```   485
```
```   486 lemma PiE_eq_iff:
```
```   487   "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 = {}))"
```
```   488 proof (intro iffI disjCI)
```
```   489   assume eq[simp]: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
```
```   490   assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
```
```   491   then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})"
```
```   492     using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by auto
```
```   493   with PiE_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i"
```
```   494     by auto
```
```   495 next
```
```   496   assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})"
```
```   497   then show "Pi\<^sub>E I F = Pi\<^sub>E I F'"
```
```   498     using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by (auto simp: PiE_def)
```
```   499 qed
```
```   500
```
```   501 lemma extensional_funcset_fun_upd_restricts_rangeI:
```
```   502   "\<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})"
```
```   503   unfolding extensional_funcset_def extensional_def
```
```   504   apply auto
```
```   505   apply (case_tac "x = xa")
```
```   506   apply auto
```
```   507   done
```
```   508
```
```   509 lemma extensional_funcset_fun_upd_extends_rangeI:
```
```   510   assumes "a \<in> T" "f \<in> S \<rightarrow>\<^sub>E (T - {a})"
```
```   511   shows "f(x := a) \<in> insert x S \<rightarrow>\<^sub>E  T"
```
```   512   using assms unfolding extensional_funcset_def extensional_def by auto
```
```   513
```
```   514
```
```   515 subsubsection \<open>Injective Extensional Function Spaces\<close>
```
```   516
```
```   517 lemma extensional_funcset_fun_upd_inj_onI:
```
```   518   assumes "f \<in> S \<rightarrow>\<^sub>E (T - {a})"
```
```   519     and "inj_on f S"
```
```   520   shows "inj_on (f(x := a)) S"
```
```   521   using assms
```
```   522   unfolding extensional_funcset_def by (auto intro!: inj_on_fun_updI)
```
```   523
```
```   524 lemma extensional_funcset_extend_domain_inj_on_eq:
```
```   525   assumes "x \<notin> S"
```
```   526   shows "{f. f \<in> (insert x S) \<rightarrow>\<^sub>E T \<and> inj_on f (insert x S)} =
```
```   527     (\<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}"
```
```   528   using assms
```
```   529   apply (auto del: PiE_I PiE_E)
```
```   530   apply (auto intro: extensional_funcset_fun_upd_inj_onI
```
```   531     extensional_funcset_fun_upd_extends_rangeI del: PiE_I PiE_E)
```
```   532   apply (auto simp add: image_iff inj_on_def)
```
```   533   apply (rule_tac x="xa x" in exI)
```
```   534   apply (auto intro: PiE_mem del: PiE_I PiE_E)
```
```   535   apply (rule_tac x="xa(x := undefined)" in exI)
```
```   536   apply (auto intro!: extensional_funcset_fun_upd_restricts_rangeI)
```
```   537   apply (auto dest!: PiE_mem split: split_if_asm)
```
```   538   done
```
```   539
```
```   540 lemma extensional_funcset_extend_domain_inj_onI:
```
```   541   assumes "x \<notin> S"
```
```   542   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}"
```
```   543   using assms
```
```   544   apply (auto intro!: inj_onI)
```
```   545   apply (metis fun_upd_same)
```
```   546   apply (metis assms PiE_arb fun_upd_triv fun_upd_upd)
```
```   547   done
```
```   548
```
```   549
```
```   550 subsubsection \<open>Cardinality\<close>
```
```   551
```
```   552 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)"
```
```   553   by (induct S arbitrary: T rule: finite_induct) (simp_all add: PiE_insert_eq)
```
```   554
```
```   555 lemma inj_combinator: "x \<notin> S \<Longrightarrow> inj_on (\<lambda>(y, g). g(x := y)) (T x \<times> Pi\<^sub>E S T)"
```
```   556 proof (safe intro!: inj_onI ext)
```
```   557   fix f y g z
```
```   558   assume "x \<notin> S"
```
```   559   assume fg: "f \<in> Pi\<^sub>E S T" "g \<in> Pi\<^sub>E S T"
```
```   560   assume "f(x := y) = g(x := z)"
```
```   561   then have *: "\<And>i. (f(x := y)) i = (g(x := z)) i"
```
```   562     unfolding fun_eq_iff by auto
```
```   563   from this[of x] show "y = z" by simp
```
```   564   fix i from *[of i] \<open>x \<notin> S\<close> fg show "f i = g i"
```
```   565     by (auto split: split_if_asm simp: PiE_def extensional_def)
```
```   566 qed
```
```   567
```
```   568 lemma card_PiE: "finite S \<Longrightarrow> card (\<Pi>\<^sub>E i \<in> S. T i) = (\<Prod> i\<in>S. card (T i))"
```
```   569 proof (induct rule: finite_induct)
```
```   570   case empty
```
```   571   then show ?case by auto
```
```   572 next
```
```   573   case (insert x S)
```
```   574   then show ?case
```
```   575     by (simp add: PiE_insert_eq inj_combinator card_image card_cartesian_product)
```
```   576 qed
```
```   577
```
```   578 end
```