src/HOL/Library/FuncSet.thy
 author wenzelm Tue Aug 13 16:25:47 2013 +0200 (2013-08-13) changeset 53015 a1119cf551e8 parent 50123 69b35a75caf3 child 53381 355a4cac5440 permissions -rw-r--r--
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
```     1 (*  Title:      HOL/Library/FuncSet.thy
```
```     2     Author:     Florian Kammueller and Lawrence C Paulson, Lukas Bulwahn
```
```     3 *)
```
```     4
```
```     5 header {* Pi and Function Sets *}
```
```     6
```
```     7 theory FuncSet
```
```     8 imports Hilbert_Choice Main
```
```     9 begin
```
```    10
```
```    11 definition
```
```    12   Pi :: "['a set, 'a => 'b set] => ('a => 'b) set" where
```
```    13   "Pi A B = {f. \<forall>x. x \<in> A --> f x \<in> B x}"
```
```    14
```
```    15 definition
```
```    16   extensional :: "'a set => ('a => 'b) set" where
```
```    17   "extensional A = {f. \<forall>x. x~:A --> f x = undefined}"
```
```    18
```
```    19 definition
```
```    20   "restrict" :: "['a => 'b, 'a set] => ('a => 'b)" where
```
```    21   "restrict f A = (%x. if x \<in> A then f x else undefined)"
```
```    22
```
```    23 abbreviation
```
```    24   funcset :: "['a set, 'b set] => ('a => 'b) set"
```
```    25     (infixr "->" 60) where
```
```    26   "A -> B == Pi A (%_. B)"
```
```    27
```
```    28 notation (xsymbols)
```
```    29   funcset  (infixr "\<rightarrow>" 60)
```
```    30
```
```    31 syntax
```
```    32   "_Pi"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI _:_./ _)" 10)
```
```    33   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3%_:_./ _)" [0,0,3] 3)
```
```    34
```
```    35 syntax (xsymbols)
```
```    36   "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
```
```    37   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
```
```    38
```
```    39 syntax (HTML output)
```
```    40   "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
```
```    41   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
```
```    42
```
```    43 translations
```
```    44   "PI x:A. B" == "CONST Pi A (%x. B)"
```
```    45   "%x:A. f" == "CONST restrict (%x. f) A"
```
```    46
```
```    47 definition
```
```    48   "compose" :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)" where
```
```    49   "compose A g f = (\<lambda>x\<in>A. g (f x))"
```
```    50
```
```    51
```
```    52 subsection{*Basic Properties of @{term Pi}*}
```
```    53
```
```    54 lemma Pi_I[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
```
```    55   by (simp add: Pi_def)
```
```    56
```
```    57 lemma Pi_I'[simp]: "(!!x. x : A --> f x : B x) ==> f : Pi A B"
```
```    58 by(simp add:Pi_def)
```
```    59
```
```    60 lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
```
```    61   by (simp add: Pi_def)
```
```    62
```
```    63 lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
```
```    64   by (simp add: Pi_def)
```
```    65
```
```    66 lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
```
```    67   unfolding Pi_def by auto
```
```    68
```
```    69 lemma PiE [elim]:
```
```    70   "f : Pi A B ==> (f x : B x ==> Q) ==> (x ~: A ==> Q) ==> Q"
```
```    71 by(auto simp: Pi_def)
```
```    72
```
```    73 lemma Pi_cong:
```
```    74   "(\<And> w. w \<in> A \<Longrightarrow> f w = g w) \<Longrightarrow> f \<in> Pi A B \<longleftrightarrow> g \<in> Pi A B"
```
```    75   by (auto simp: Pi_def)
```
```    76
```
```    77 lemma funcset_id [simp]: "(\<lambda>x. x) \<in> A \<rightarrow> A"
```
```    78   by auto
```
```    79
```
```    80 lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
```
```    81   by (simp add: Pi_def)
```
```    82
```
```    83 lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B"
```
```    84   by auto
```
```    85
```
```    86 lemma image_subset_iff_funcset: "F ` A \<subseteq> B \<longleftrightarrow> F \<in> A \<rightarrow> B"
```
```    87   by auto
```
```    88
```
```    89 lemma Pi_eq_empty[simp]: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B x = {})"
```
```    90 apply (simp add: Pi_def, auto)
```
```    91 txt{*Converse direction requires Axiom of Choice to exhibit a function
```
```    92 picking an element from each non-empty @{term "B x"}*}
```
```    93 apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec, auto)
```
```    94 apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto)
```
```    95 done
```
```    96
```
```    97 lemma Pi_empty [simp]: "Pi {} B = UNIV"
```
```    98 by (simp add: Pi_def)
```
```    99
```
```   100 lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)"
```
```   101   by auto
```
```   102
```
```   103 lemma Pi_UN:
```
```   104   fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
```
```   105   assumes "finite I" and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
```
```   106   shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)"
```
```   107 proof (intro set_eqI iffI)
```
```   108   fix f assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)"
```
```   109   then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i" by auto
```
```   110   from bchoice[OF this] obtain n where n: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> (A (n i) i)" by auto
```
```   111   obtain k where k: "\<And>i. i \<in> I \<Longrightarrow> n i \<le> k"
```
```   112     using `finite I` finite_nat_set_iff_bounded_le[of "n`I"] by auto
```
```   113   have "f \<in> Pi I (A k)"
```
```   114   proof (intro Pi_I)
```
```   115     fix i assume "i \<in> I"
```
```   116     from mono[OF this, of "n i" k] k[OF this] n[OF this]
```
```   117     show "f i \<in> A k i" by auto
```
```   118   qed
```
```   119   then show "f \<in> (\<Union>n. Pi I (A n))" by auto
```
```   120 qed auto
```
```   121
```
```   122 lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
```
```   123 by (simp add: Pi_def)
```
```   124
```
```   125 text{*Covariance of Pi-sets in their second argument*}
```
```   126 lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"
```
```   127 by auto
```
```   128
```
```   129 text{*Contravariance of Pi-sets in their first argument*}
```
```   130 lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"
```
```   131 by auto
```
```   132
```
```   133 lemma prod_final:
```
```   134   assumes 1: "fst \<circ> f \<in> Pi A B" and 2: "snd \<circ> f \<in> Pi A C"
```
```   135   shows "f \<in> (\<Pi> z \<in> A. B z \<times> C z)"
```
```   136 proof (rule Pi_I)
```
```   137   fix z
```
```   138   assume z: "z \<in> A"
```
```   139   have "f z = (fst (f z), snd (f z))"
```
```   140     by simp
```
```   141   also have "...  \<in> B z \<times> C z"
```
```   142     by (metis SigmaI PiE o_apply 1 2 z)
```
```   143   finally show "f z \<in> B z \<times> C z" .
```
```   144 qed
```
```   145
```
```   146 lemma Pi_split_domain[simp]: "x \<in> Pi (I \<union> J) X \<longleftrightarrow> x \<in> Pi I X \<and> x \<in> Pi J X"
```
```   147   by (auto simp: Pi_def)
```
```   148
```
```   149 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"
```
```   150   by (auto simp: Pi_def)
```
```   151
```
```   152 lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B"
```
```   153   by (auto simp: Pi_def)
```
```   154
```
```   155 lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
```
```   156   by (auto simp: Pi_def)
```
```   157
```
```   158 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"
```
```   159   apply auto
```
```   160   apply (drule_tac x=x in Pi_mem)
```
```   161   apply (simp_all split: split_if_asm)
```
```   162   apply (drule_tac x=i in Pi_mem)
```
```   163   apply (auto dest!: Pi_mem)
```
```   164   done
```
```   165
```
```   166 subsection{*Composition With a Restricted Domain: @{term compose}*}
```
```   167
```
```   168 lemma funcset_compose:
```
```   169   "[| f \<in> A -> B; g \<in> B -> C |]==> compose A g f \<in> A -> C"
```
```   170 by (simp add: Pi_def compose_def restrict_def)
```
```   171
```
```   172 lemma compose_assoc:
```
```   173     "[| f \<in> A -> B; g \<in> B -> C; h \<in> C -> D |]
```
```   174       ==> compose A h (compose A g f) = compose A (compose B h g) f"
```
```   175 by (simp add: fun_eq_iff Pi_def compose_def restrict_def)
```
```   176
```
```   177 lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"
```
```   178 by (simp add: compose_def restrict_def)
```
```   179
```
```   180 lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
```
```   181   by (auto simp add: image_def compose_eq)
```
```   182
```
```   183
```
```   184 subsection{*Bounded Abstraction: @{term restrict}*}
```
```   185
```
```   186 lemma restrict_in_funcset: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A -> B"
```
```   187   by (simp add: Pi_def restrict_def)
```
```   188
```
```   189 lemma restrictI[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
```
```   190   by (simp add: Pi_def restrict_def)
```
```   191
```
```   192 lemma restrict_apply [simp]:
```
```   193     "(\<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_ext:
```
```   197     "(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
```
```   198   by (simp add: fun_eq_iff Pi_def restrict_def)
```
```   199
```
```   200 lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
```
```   201   by (simp add: inj_on_def restrict_def)
```
```   202
```
```   203 lemma Id_compose:
```
```   204     "[|f \<in> A -> B;  f \<in> extensional A|] ==> compose A (\<lambda>y\<in>B. y) f = f"
```
```   205   by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
```
```   206
```
```   207 lemma compose_Id:
```
```   208     "[|g \<in> A -> B;  g \<in> extensional A|] ==> compose A g (\<lambda>x\<in>A. x) = g"
```
```   209   by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
```
```   210
```
```   211 lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
```
```   212   by (auto simp add: restrict_def)
```
```   213
```
```   214 lemma restrict_restrict[simp]: "restrict (restrict f A) B = restrict f (A \<inter> B)"
```
```   215   unfolding restrict_def by (simp add: fun_eq_iff)
```
```   216
```
```   217 lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I"
```
```   218   by (auto simp: restrict_def)
```
```   219
```
```   220 lemma restrict_upd[simp]:
```
```   221   "i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"
```
```   222   by (auto simp: fun_eq_iff)
```
```   223
```
```   224 lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A"
```
```   225   by (auto simp: restrict_def Pi_def)
```
```   226
```
```   227
```
```   228 subsection{*Bijections Between Sets*}
```
```   229
```
```   230 text{*The definition of @{const bij_betw} is in @{text "Fun.thy"}, but most of
```
```   231 the theorems belong here, or need at least @{term Hilbert_Choice}.*}
```
```   232
```
```   233 lemma bij_betwI:
```
```   234 assumes "f \<in> A \<rightarrow> B" and "g \<in> B \<rightarrow> A"
```
```   235     and g_f: "\<And>x. x\<in>A \<Longrightarrow> g (f x) = x" and f_g: "\<And>y. y\<in>B \<Longrightarrow> f (g y) = y"
```
```   236 shows "bij_betw f A B"
```
```   237 unfolding bij_betw_def
```
```   238 proof
```
```   239   show "inj_on f A" by (metis g_f inj_on_def)
```
```   240 next
```
```   241   have "f ` A \<subseteq> B" using `f \<in> A \<rightarrow> B` by auto
```
```   242   moreover
```
```   243   have "B \<subseteq> f ` A" by auto (metis Pi_mem `g \<in> B \<rightarrow> A` f_g image_iff)
```
```   244   ultimately show "f ` A = B" by blast
```
```   245 qed
```
```   246
```
```   247 lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
```
```   248 by (auto simp add: bij_betw_def)
```
```   249
```
```   250 lemma inj_on_compose:
```
```   251   "[| bij_betw f A B; inj_on g B |] ==> 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:
```
```   255   "[| bij_betw f A B; bij_betw g B C |] ==> bij_betw (compose A g f) A C"
```
```   256 apply (simp add: bij_betw_def compose_eq inj_on_compose)
```
```   257 apply (auto simp add: compose_def image_def)
```
```   258 done
```
```   259
```
```   260 lemma bij_betw_restrict_eq [simp]:
```
```   261   "bij_betw (restrict f A) A B = bij_betw f A B"
```
```   262 by (simp add: bij_betw_def)
```
```   263
```
```   264
```
```   265 subsection{*Extensionality*}
```
```   266
```
```   267 lemma extensional_empty[simp]: "extensional {} = {\<lambda>x. undefined}"
```
```   268   unfolding extensional_def by auto
```
```   269
```
```   270 lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = undefined"
```
```   271 by (simp add: extensional_def)
```
```   272
```
```   273 lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
```
```   274 by (simp add: restrict_def extensional_def)
```
```   275
```
```   276 lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
```
```   277 by (simp add: compose_def)
```
```   278
```
```   279 lemma extensionalityI:
```
```   280   "[| f \<in> extensional A; g \<in> extensional A;
```
```   281       !!x. x\<in>A ==> f x = g x |] ==> f = g"
```
```   282 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 ==> (\<lambda>x\<in>B. inv_into A f x) : B -> A"
```
```   291 by (unfold inv_into_def) (fast intro: someI2)
```
```   292
```
```   293 lemma compose_inv_into_id:
```
```   294   "bij_betw f A B ==> compose A (\<lambda>y\<in>B. inv_into A f y) f = (\<lambda>x\<in>A. x)"
```
```   295 apply (simp add: bij_betw_def compose_def)
```
```   296 apply (rule restrict_ext, auto)
```
```   297 done
```
```   298
```
```   299 lemma compose_id_inv_into:
```
```   300   "f ` A = B ==> compose B f (\<lambda>y\<in>B. inv_into A f y) = (\<lambda>x\<in>B. x)"
```
```   301 apply (simp add: compose_def)
```
```   302 apply (rule restrict_ext)
```
```   303 apply (simp add: f_inv_into_f)
```
```   304 done
```
```   305
```
```   306 lemma extensional_insert[intro, simp]:
```
```   307   assumes "a \<in> extensional (insert i I)"
```
```   308   shows "a(i := b) \<in> extensional (insert i I)"
```
```   309   using assms unfolding extensional_def by auto
```
```   310
```
```   311 lemma extensional_Int[simp]:
```
```   312   "extensional I \<inter> extensional I' = extensional (I \<inter> I')"
```
```   313   unfolding extensional_def by auto
```
```   314
```
```   315 lemma extensional_UNIV[simp]: "extensional UNIV = UNIV"
```
```   316   by (auto simp: extensional_def)
```
```   317
```
```   318 lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B"
```
```   319   unfolding restrict_def extensional_def by auto
```
```   320
```
```   321 lemma extensional_insert_undefined[intro, simp]:
```
```   322   "a \<in> extensional (insert i I) \<Longrightarrow> a(i := undefined) \<in> extensional I"
```
```   323   unfolding extensional_def by auto
```
```   324
```
```   325 lemma extensional_insert_cancel[intro, simp]:
```
```   326   "a \<in> extensional I \<Longrightarrow> a \<in> extensional (insert i I)"
```
```   327   unfolding extensional_def by auto
```
```   328
```
```   329
```
```   330 subsection{*Cardinality*}
```
```   331
```
```   332 lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
```
```   333 by (rule card_inj_on_le) auto
```
```   334
```
```   335 lemma card_bij:
```
```   336   "[|f \<in> A\<rightarrow>B; inj_on f A;
```
```   337      g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
```
```   338 by (blast intro: card_inj order_antisym)
```
```   339
```
```   340 subsection {* Extensional Function Spaces *}
```
```   341
```
```   342 definition PiE :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set" where
```
```   343   "PiE S T = Pi S T \<inter> extensional S"
```
```   344
```
```   345 abbreviation "Pi\<^sub>E A B \<equiv> PiE A B"
```
```   346
```
```   347 syntax "_PiE"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PIE _:_./ _)" 10)
```
```   348
```
```   349 syntax (xsymbols) "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^sub>E _\<in>_./ _)" 10)
```
```   350
```
```   351 syntax (HTML output) "_PiE" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi>\<^sub>E _\<in>_./ _)" 10)
```
```   352
```
```   353 translations "PIE x:A. B" == "CONST Pi\<^sub>E A (%x. B)"
```
```   354
```
```   355 abbreviation extensional_funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr "->\<^sub>E" 60) where
```
```   356   "A ->\<^sub>E B \<equiv> (\<Pi>\<^sub>E i\<in>A. B)"
```
```   357
```
```   358 notation (xsymbols)
```
```   359   extensional_funcset  (infixr "\<rightarrow>\<^sub>E" 60)
```
```   360
```
```   361 lemma extensional_funcset_def: "extensional_funcset S T = (S -> T) \<inter> extensional S"
```
```   362   by (simp add: PiE_def)
```
```   363
```
```   364 lemma PiE_empty_domain[simp]: "PiE {} T = {%x. undefined}"
```
```   365   unfolding PiE_def by simp
```
```   366
```
```   367 lemma PiE_empty_range[simp]: "i \<in> I \<Longrightarrow> F i = {} \<Longrightarrow> (PIE i:I. F i) = {}"
```
```   368   unfolding PiE_def by auto
```
```   369
```
```   370 lemma PiE_eq_empty_iff:
```
```   371   "Pi\<^sub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})"
```
```   372 proof
```
```   373   assume "Pi\<^sub>E I F = {}"
```
```   374   show "\<exists>i\<in>I. F i = {}"
```
```   375   proof (rule ccontr)
```
```   376     assume "\<not> ?thesis"
```
```   377     then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)" by auto
```
```   378     from choice[OF this] guess f ..
```
```   379     then have "f \<in> Pi\<^sub>E I F" by (auto simp: extensional_def PiE_def)
```
```   380     with `Pi\<^sub>E I F = {}` show False by auto
```
```   381   qed
```
```   382 qed (auto simp: PiE_def)
```
```   383
```
```   384 lemma PiE_arb: "f \<in> PiE 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> PiE 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> PiE S T \<Longrightarrow> f(x := y) \<in> PiE (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> PiE (insert x S) T \<Longrightarrow> f(x := undefined) \<in> PiE S T"
```
```   394   unfolding PiE_def extensional_def by auto
```
```   395
```
```   396 lemma PiE_insert_eq:
```
```   397   assumes "x \<notin> S"
```
```   398   shows "PiE (insert x S) T = (\<lambda>(y, g). g(x := y)) ` (T x \<times> PiE S T)"
```
```   399 proof -
```
```   400   {
```
```   401     fix f assume "f \<in> PiE (insert x S) T"
```
```   402     with assms have "f \<in> (\<lambda>(y, g). g(x := y)) ` (T x \<times> PiE S T)"
```
```   403       by (auto intro!: image_eqI[where x="(f x, f(x := undefined))"] intro: fun_upd_in_PiE PiE_mem)
```
```   404   }
```
```   405   then show ?thesis using assms by (auto intro: PiE_fun_upd)
```
```   406 qed
```
```   407
```
```   408 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)"
```
```   409   by (auto simp: PiE_def)
```
```   410
```
```   411 lemma PiE_cong:
```
```   412   "(\<And>i. i\<in>I \<Longrightarrow> A i = B i) \<Longrightarrow> Pi\<^sub>E I A = Pi\<^sub>E I B"
```
```   413   unfolding PiE_def by (auto simp: Pi_cong)
```
```   414
```
```   415 lemma PiE_E [elim]:
```
```   416   "f \<in> PiE A B \<Longrightarrow> (x \<in> A \<Longrightarrow> f x \<in> B x \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> f x = undefined \<Longrightarrow> Q) \<Longrightarrow> Q"
```
```   417 by(auto simp: Pi_def PiE_def extensional_def)
```
```   418
```
```   419 lemma PiE_I[intro!]: "(\<And>x. x \<in> A ==> f x \<in> B x) \<Longrightarrow> (\<And>x. x \<notin> A \<Longrightarrow> f x = undefined) \<Longrightarrow> f \<in> PiE A B"
```
```   420   by (simp add: PiE_def extensional_def)
```
```   421
```
```   422 lemma PiE_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> PiE A B \<subseteq> PiE A C"
```
```   423   by auto
```
```   424
```
```   425 lemma PiE_iff: "f \<in> PiE I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i) \<and> f \<in> extensional I"
```
```   426   by (simp add: PiE_def Pi_iff)
```
```   427
```
```   428 lemma PiE_restrict[simp]:  "f \<in> PiE A B \<Longrightarrow> restrict f A = f"
```
```   429   by (simp add: extensional_restrict PiE_def)
```
```   430
```
```   431 lemma restrict_PiE[simp]: "restrict f I \<in> PiE I S \<longleftrightarrow> f \<in> Pi I S"
```
```   432   by (auto simp: PiE_iff)
```
```   433
```
```   434 lemma PiE_eq_subset:
```
```   435   assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
```
```   436   assumes eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'" and "i \<in> I"
```
```   437   shows "F i \<subseteq> F' i"
```
```   438 proof
```
```   439   fix x assume "x \<in> F i"
```
```   440   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))" by auto
```
```   441   from choice[OF this] guess f .. note f = this
```
```   442   then have "f \<in> Pi\<^sub>E I F" by (auto simp: extensional_def PiE_def)
```
```   443   then have "f \<in> Pi\<^sub>E I F'" using assms by simp
```
```   444   then show "x \<in> F' i" using f `i \<in> I` by (auto simp: PiE_def)
```
```   445 qed
```
```   446
```
```   447 lemma PiE_eq_iff_not_empty:
```
```   448   assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
```
```   449   shows "Pi\<^sub>E I F = Pi\<^sub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)"
```
```   450 proof (intro iffI ballI)
```
```   451   fix i assume eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'" and i: "i \<in> I"
```
```   452   show "F i = F' i"
```
```   453     using PiE_eq_subset[of I F F', OF ne eq i]
```
```   454     using PiE_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]
```
```   455     by auto
```
```   456 qed (auto simp: PiE_def)
```
```   457
```
```   458 lemma PiE_eq_iff:
```
```   459   "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 = {}))"
```
```   460 proof (intro iffI disjCI)
```
```   461   assume eq[simp]: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
```
```   462   assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
```
```   463   then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})"
```
```   464     using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by auto
```
```   465   with PiE_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i" by auto
```
```   466 next
```
```   467   assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})"
```
```   468   then show "Pi\<^sub>E I F = Pi\<^sub>E I F'"
```
```   469     using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by (auto simp: PiE_def)
```
```   470 qed
```
```   471
```
```   472 lemma extensional_funcset_fun_upd_restricts_rangeI:
```
```   473   "\<forall>y \<in> S. f x \<noteq> f y \<Longrightarrow> f : (insert x S) \<rightarrow>\<^sub>E T ==> f(x := undefined) : S \<rightarrow>\<^sub>E (T - {f x})"
```
```   474   unfolding extensional_funcset_def extensional_def
```
```   475   apply auto
```
```   476   apply (case_tac "x = xa")
```
```   477   apply auto
```
```   478   done
```
```   479
```
```   480 lemma extensional_funcset_fun_upd_extends_rangeI:
```
```   481   assumes "a \<in> T" "f \<in> S \<rightarrow>\<^sub>E (T - {a})"
```
```   482   shows "f(x := a) \<in> (insert x S) \<rightarrow>\<^sub>E  T"
```
```   483   using assms unfolding extensional_funcset_def extensional_def by auto
```
```   484
```
```   485 subsubsection {* Injective Extensional Function Spaces *}
```
```   486
```
```   487 lemma extensional_funcset_fun_upd_inj_onI:
```
```   488   assumes "f \<in> S \<rightarrow>\<^sub>E (T - {a})" "inj_on f S"
```
```   489   shows "inj_on (f(x := a)) S"
```
```   490   using assms unfolding extensional_funcset_def by (auto intro!: inj_on_fun_updI)
```
```   491
```
```   492 lemma extensional_funcset_extend_domain_inj_on_eq:
```
```   493   assumes "x \<notin> S"
```
```   494   shows"{f. f \<in> (insert x S) \<rightarrow>\<^sub>E T \<and> inj_on f (insert x S)} =
```
```   495     (%(y, g). g(x:=y)) ` {(y, g). y \<in> T \<and> g \<in> S \<rightarrow>\<^sub>E (T - {y}) \<and> inj_on g S}"
```
```   496 proof -
```
```   497   from assms show ?thesis
```
```   498     apply (auto del: PiE_I PiE_E)
```
```   499     apply (auto intro: extensional_funcset_fun_upd_inj_onI extensional_funcset_fun_upd_extends_rangeI del: PiE_I PiE_E)
```
```   500     apply (auto simp add: image_iff inj_on_def)
```
```   501     apply (rule_tac x="xa x" in exI)
```
```   502     apply (auto intro: PiE_mem del: PiE_I PiE_E)
```
```   503     apply (rule_tac x="xa(x := undefined)" in exI)
```
```   504     apply (auto intro!: extensional_funcset_fun_upd_restricts_rangeI)
```
```   505     apply (auto dest!: PiE_mem split: split_if_asm)
```
```   506     done
```
```   507 qed
```
```   508
```
```   509 lemma extensional_funcset_extend_domain_inj_onI:
```
```   510   assumes "x \<notin> S"
```
```   511   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}"
```
```   512 proof -
```
```   513   from assms show ?thesis
```
```   514     apply (auto intro!: inj_onI)
```
```   515     apply (metis fun_upd_same)
```
```   516     by (metis assms PiE_arb fun_upd_triv fun_upd_upd)
```
```   517 qed
```
```   518
```
```   519
```
```   520 subsubsection {* Cardinality *}
```
```   521
```
```   522 lemma finite_PiE: "finite S \<Longrightarrow> (\<And>i. i \<in> S \<Longrightarrow> finite (T i)) \<Longrightarrow> finite (PIE i : S. T i)"
```
```   523   by (induct S arbitrary: T rule: finite_induct) (simp_all add: PiE_insert_eq)
```
```   524
```
```   525 lemma inj_combinator: "x \<notin> S \<Longrightarrow> inj_on (\<lambda>(y, g). g(x := y)) (T x \<times> Pi\<^sub>E S T)"
```
```   526 proof (safe intro!: inj_onI ext)
```
```   527   fix f y g z assume "x \<notin> S" and fg: "f \<in> Pi\<^sub>E S T" "g \<in> Pi\<^sub>E S T"
```
```   528   assume "f(x := y) = g(x := z)"
```
```   529   then have *: "\<And>i. (f(x := y)) i = (g(x := z)) i"
```
```   530     unfolding fun_eq_iff by auto
```
```   531   from this[of x] show "y = z" by simp
```
```   532   fix i from *[of i] `x \<notin> S` fg show "f i = g i"
```
```   533     by (auto split: split_if_asm simp: PiE_def extensional_def)
```
```   534 qed
```
```   535
```
```   536 lemma card_PiE:
```
```   537   "finite S \<Longrightarrow> card (PIE i : S. T i) = (\<Prod> i\<in>S. card (T i))"
```
```   538 proof (induct rule: finite_induct)
```
```   539   case empty then show ?case by auto
```
```   540 next
```
```   541   case (insert x S) then show ?case
```
```   542     by (simp add: PiE_insert_eq inj_combinator card_image card_cartesian_product)
```
```   543 qed
```
```   544
```
```   545 end
```