src/HOL/Library/FuncSet.thy
author wenzelm
Mon, 11 Sep 2023 19:30:48 +0200
changeset 78659 b5f3d1051b13
parent 78248 740b23f1138a
child 80768 c7723cc15de8
permissions -rw-r--r--
tuned;
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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(*  Title:      HOL/Library/FuncSet.thy
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    Author:     Florian Kammueller and Lawrence C Paulson, Lukas Bulwahn
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*)
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section \<open>Pi and Function Sets\<close>
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theory FuncSet
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  imports Main
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  abbrevs PiE = "Pi\<^sub>E"
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    and PIE = "\<Pi>\<^sub>E"
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begin
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definition Pi :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set"
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  where "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}"
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definition extensional :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) set"
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  where "extensional A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = undefined}"
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definition "restrict" :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
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  where "restrict f A = (\<lambda>x. if x \<in> A then f x else undefined)"
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abbreviation funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  (infixr "\<rightarrow>" 60)
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  where "A \<rightarrow> B \<equiv> Pi A (\<lambda>_. B)"
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syntax
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  "_Pi" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
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  "_lam" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
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translations
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  "\<Pi> x\<in>A. B" \<rightleftharpoons> "CONST Pi A (\<lambda>x. B)"
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  "\<lambda>x\<in>A. f" \<rightleftharpoons> "CONST restrict (\<lambda>x. f) A"
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definition "compose" :: "'a set \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'c)"
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  where "compose A g f = (\<lambda>x\<in>A. g (f x))"
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subsection \<open>Basic Properties of \<^term>\<open>Pi\<close>\<close>
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lemma Pi_I[intro!]: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi A B"
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  by (simp add: Pi_def)
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lemma Pi_I'[simp]: "(\<And>x. x \<in> A \<longrightarrow> f x \<in> B x) \<Longrightarrow> f \<in> Pi A B"
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  by (simp add:Pi_def)
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lemma funcsetI: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> f \<in> A \<rightarrow> B"
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  by (simp add: Pi_def)
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lemma Pi_mem: "f \<in> Pi A B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B x"
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  by (simp add: Pi_def)
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lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
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  unfolding Pi_def by auto
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lemma PiE [elim]: "f \<in> Pi A B \<Longrightarrow> (f x \<in> B x \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> Q) \<Longrightarrow> Q"
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  by (auto simp: Pi_def)
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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"
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  by (auto simp: Pi_def)
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    58
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lemma funcset_id [simp]: "(\<lambda>x. x) \<in> A \<rightarrow> A"
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  by auto
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lemma funcset_mem: "f \<in> A \<rightarrow> B \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<in> B"
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    63
  by (simp add: Pi_def)
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lemma funcset_image: "f \<in> A \<rightarrow> B \<Longrightarrow> f ` A \<subseteq> B"
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  by auto
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de19856feb54 move theorems to be more generally useable
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lemma image_subset_iff_funcset: "F ` A \<subseteq> B \<longleftrightarrow> F \<in> A \<rightarrow> B"
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    69
  by auto
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    70
71258
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lemma funcset_to_empty_iff: "A \<rightarrow> {} = (if A={} then UNIV else {})"
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    72
  by auto
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    73
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lemma Pi_eq_empty[simp]: "(\<Pi> x \<in> A. B x) = {} \<longleftrightarrow> (\<exists>x\<in>A. B x = {})"
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proof -
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    76
  have "\<exists>x\<in>A. B x = {}" if "\<And>f. \<exists>y. y \<in> A \<and> f y \<notin> B y"
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    77
    using that [of "\<lambda>u. SOME y. y \<in> B u"] some_in_eq by blast
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  then show ?thesis
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    by force
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    80
qed
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lemma Pi_empty [simp]: "Pi {} B = UNIV"
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  by (simp add: Pi_def)
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    84
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lemma Pi_Int: "Pi I E \<inter> Pi I F = (\<Pi> i\<in>I. E i \<inter> F i)"
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    86
  by auto
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    87
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lemma Pi_UN:
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  fixes A :: "nat \<Rightarrow> 'i \<Rightarrow> 'a set"
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  assumes "finite I"
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    91
    and mono: "\<And>i n m. i \<in> I \<Longrightarrow> n \<le> m \<Longrightarrow> A n i \<subseteq> A m i"
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    92
  shows "(\<Union>n. Pi I (A n)) = (\<Pi> i\<in>I. \<Union>n. A n i)"
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    93
proof (intro set_eqI iffI)
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    94
  fix f
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    95
  assume "f \<in> (\<Pi> i\<in>I. \<Union>n. A n i)"
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    96
  then have "\<forall>i\<in>I. \<exists>n. f i \<in> A n i"
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    97
    by auto
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    98
  from bchoice[OF this] obtain n where n: "f i \<in> A (n i) i" if "i \<in> I" for i
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    99
    by auto
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   100
  obtain k where k: "n i \<le> k" if "i \<in> I" for i
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   101
    using \<open>finite I\<close> finite_nat_set_iff_bounded_le[of "n`I"] by auto
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   102
  have "f \<in> Pi I (A k)"
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   103
  proof (intro Pi_I)
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   104
    fix i
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   105
    assume "i \<in> I"
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   106
    from mono[OF this, of "n i" k] k[OF this] n[OF this]
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   107
    show "f i \<in> A k i" by auto
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   108
  qed
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   109
  then show "f \<in> (\<Union>n. Pi I (A n))"
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   110
    by auto
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   111
qed auto
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hoelzl
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   112
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   113
lemma Pi_UNIV [simp]: "A \<rightarrow> UNIV = UNIV"
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   114
  by (simp add: Pi_def)
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hoelzl
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diff changeset
   115
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   116
text \<open>Covariance of Pi-sets in their second argument\<close>
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   117
lemma Pi_mono: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x) \<Longrightarrow> Pi A B \<subseteq> Pi A C"
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   118
  by auto
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parents:
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   119
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   120
text \<open>Contravariance of Pi-sets in their first argument\<close>
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   121
lemma Pi_anti_mono: "A' \<subseteq> A \<Longrightarrow> Pi A B \<subseteq> Pi A' B"
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wenzelm
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   122
  by auto
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   123
33271
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paulson
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   124
lemma prod_final:
58783
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   125
  assumes 1: "fst \<circ> f \<in> Pi A B"
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   126
    and 2: "snd \<circ> f \<in> Pi A C"
33271
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paulson
parents: 33057
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   127
  shows "f \<in> (\<Pi> z \<in> A. B z \<times> C z)"
58783
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   128
proof (rule Pi_I)
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   129
  fix z
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   130
  assume z: "z \<in> A"
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   131
  have "f z = (fst (f z), snd (f z))"
33271
7be66dee1a5a New theory Probability, which contains a development of measure theory
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   132
    by simp
58783
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   133
  also have "\<dots> \<in> B z \<times> C z"
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diff changeset
   134
    by (metis SigmaI PiE o_apply 1 2 z)
33271
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 33057
diff changeset
   135
  finally show "f z \<in> B z \<times> C z" .
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 33057
diff changeset
   136
qed
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 33057
diff changeset
   137
50123
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diff changeset
   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"
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hoelzl
parents: 50104
diff changeset
   139
  by (auto simp: Pi_def)
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   140
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
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   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"
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  by (auto simp: Pi_def)
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   143
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lemma Pi_cancel_fupd_range[simp]: "i \<notin> I \<Longrightarrow> x \<in> Pi I (B(i := b)) \<longleftrightarrow> x \<in> Pi I B"
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   145
  by (auto simp: Pi_def)
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   146
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lemma Pi_cancel_fupd[simp]: "i \<notin> I \<Longrightarrow> x(i := a) \<in> Pi I B \<longleftrightarrow> x \<in> Pi I B"
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   148
  by (auto simp: Pi_def)
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   149
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   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"
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   151
  using mk_disjoint_insert by fastforce
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   152
740b23f1138a EXPERIMENTAL replacement of f ` A <= B by f : A -> B in Analysis
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lemma fst_Pi: "fst \<in> A \<times> B \<rightarrow> A" and snd_Pi: "snd \<in> A \<times> B \<rightarrow> B"
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   154
  by auto
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subsection \<open>Composition With a Restricted Domain: \<^term>\<open>compose\<close>\<close>
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   159
lemma funcset_compose: "f \<in> A \<rightarrow> B \<Longrightarrow> g \<in> B \<rightarrow> C \<Longrightarrow> compose A g f \<in> A \<rightarrow> C"
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   160
  by (simp add: Pi_def compose_def restrict_def)
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lemma compose_assoc:
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  assumes "f \<in> A \<rightarrow> B"
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   164
  shows "compose A h (compose A g f) = compose A (compose B h g) f"
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  using assms by (simp add: fun_eq_iff Pi_def compose_def restrict_def)
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   166
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lemma compose_eq: "x \<in> A \<Longrightarrow> compose A g f x = g (f x)"
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  by (simp add: compose_def restrict_def)
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lemma surj_compose: "f ` A = B \<Longrightarrow> g ` B = C \<Longrightarrow> compose A g f ` A = C"
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   171
  by (auto simp add: image_def compose_eq)
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   172
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   173
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   174
subsection \<open>Bounded Abstraction: \<^term>\<open>restrict\<close>\<close>
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   175
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lemma restrict_cong: "I = J \<Longrightarrow> (\<And>i. i \<in> J =simp=> f i = g i) \<Longrightarrow> restrict f I = restrict g J"
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   177
  by (auto simp: restrict_def fun_eq_iff simp_implies_def)
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   178
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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"
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  by (simp add: Pi_def restrict_def)
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lemma restrict_apply[simp]: "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else undefined)"
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  by (simp add: restrict_def)
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   185
lemma restrict_apply': "x \<in> A \<Longrightarrow> (\<lambda>y\<in>A. f y) x = f x"
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  by simp
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   187
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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)"
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   189
  by (simp add: fun_eq_iff Pi_def restrict_def)
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lemma restrict_UNIV: "restrict f UNIV = f"
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   192
  by (simp add: restrict_def)
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   193
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   194
lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A \<longleftrightarrow> inj_on f A"
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   195
  by (simp add: inj_on_def restrict_def)
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parents:
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   196
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   197
lemma inj_on_restrict_iff: "A \<subseteq> B \<Longrightarrow> inj_on (restrict f B) A \<longleftrightarrow> inj_on f A"
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parents: 73932
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   198
  by (metis inj_on_cong restrict_def subset_iff)
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   199
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lemma Id_compose: "f \<in> A \<rightarrow> B \<Longrightarrow> f \<in> extensional A \<Longrightarrow> compose A (\<lambda>y\<in>B. y) f = f"
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   201
  by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
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   202
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   203
lemma compose_Id: "g \<in> A \<rightarrow> B \<Longrightarrow> g \<in> extensional A \<Longrightarrow> compose A g (\<lambda>x\<in>A. x) = g"
39302
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   204
  by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
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   205
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   206
lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
19736
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   207
  by (auto simp add: restrict_def)
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50123
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   209
lemma restrict_restrict[simp]: "restrict (restrict f A) B = restrict f (A \<inter> B)"
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   210
  unfolding restrict_def by (simp add: fun_eq_iff)
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   211
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
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   212
lemma restrict_fupd[simp]: "i \<notin> I \<Longrightarrow> restrict (f (i := x)) I = restrict f I"
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   213
  by (auto simp: restrict_def)
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   214
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   215
lemma restrict_upd[simp]: "i \<notin> I \<Longrightarrow> (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"
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   216
  by (auto simp: fun_eq_iff)
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   217
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
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   218
lemma restrict_Pi_cancel: "restrict x I \<in> Pi I A \<longleftrightarrow> x \<in> Pi I A"
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   219
  by (auto simp: restrict_def Pi_def)
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diff changeset
   220
70063
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parents: 69939
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   221
lemma sum_restrict' [simp]: "sum' (\<lambda>i\<in>I. g i) I = sum' (\<lambda>i. g i) I"
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   222
  by (simp add: sum.G_def conj_commute cong: conj_cong)
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parents: 69939
diff changeset
   223
adaa0a6ea4fe fixes for Free_Abelian_Groups
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   224
lemma prod_restrict' [simp]: "prod' (\<lambda>i\<in>I. g i) I = prod' (\<lambda>i. g i) I"
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parents: 69939
diff changeset
   225
  by (simp add: prod.G_def conj_commute cong: conj_cong)
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parents: 69939
diff changeset
   226
14745
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   227
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subsection \<open>Bijections Between Sets\<close>
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   229
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   230
text \<open>The definition of \<^const>\<open>bij_betw\<close> is in \<open>Fun.thy\<close>, but most of
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   231
the theorems belong here, or need at least \<^term>\<open>Hilbert_Choice\<close>.\<close>
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   232
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   233
lemma bij_betwI:
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   234
  assumes "f \<in> A \<rightarrow> B"
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   235
    and "g \<in> B \<rightarrow> A"
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   236
    and g_f: "\<And>x. x\<in>A \<Longrightarrow> g (f x) = x"
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   237
    and f_g: "\<And>y. y\<in>B \<Longrightarrow> f (g y) = y"
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   238
  shows "bij_betw f A B"
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   239
  unfolding bij_betw_def
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   240
proof
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   241
  show "inj_on f A"
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   242
    by (metis g_f inj_on_def)
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   243
  have "f ` A \<subseteq> B"
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   244
    using \<open>f \<in> A \<rightarrow> B\<close> by auto
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   245
  moreover
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   246
  have "B \<subseteq> f ` A"
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diff changeset
   247
    by auto (metis Pi_mem \<open>g \<in> B \<rightarrow> A\<close> f_g image_iff)
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diff changeset
   248
  ultimately show "f ` A = B"
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   249
    by blast
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   250
qed
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   251
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   252
lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
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   253
  by (auto simp add: bij_betw_def)
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   254
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   255
lemma inj_on_compose: "bij_betw f A B \<Longrightarrow> inj_on g B \<Longrightarrow> inj_on (compose A g f) A"
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   256
  by (auto simp add: bij_betw_def inj_on_def compose_eq)
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   257
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   258
lemma bij_betw_compose: "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (compose A g f) A C"
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   259
  apply (simp add: bij_betw_def compose_eq inj_on_compose)
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   260
  apply (auto simp add: compose_def image_def)
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   261
  done
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diff changeset
   262
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   263
lemma bij_betw_restrict_eq [simp]: "bij_betw (restrict f A) A B = bij_betw f A B"
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   264
  by (simp add: bij_betw_def)
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   265
8d710bece29f more on bij_betw
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diff changeset
   266
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subsection \<open>Extensionality\<close>
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diff changeset
   268
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   269
lemma extensional_empty[simp]: "extensional {} = {\<lambda>x. undefined}"
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   270
  unfolding extensional_def by auto
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diff changeset
   271
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   272
lemma extensional_arb: "f \<in> extensional A \<Longrightarrow> x \<notin> A \<Longrightarrow> f x = undefined"
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   273
  by (simp add: extensional_def)
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diff changeset
   274
8d710bece29f more on bij_betw
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   275
lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
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   276
  by (simp add: restrict_def extensional_def)
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parents: 14762
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   277
8d710bece29f more on bij_betw
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   278
lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
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   279
  by (simp add: compose_def)
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diff changeset
   280
8d710bece29f more on bij_betw
paulson
parents: 14762
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   281
lemma extensionalityI:
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  assumes "f \<in> extensional A"
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parents: 58606
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   283
    and "g \<in> extensional A"
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parents: 58606
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   284
    and "\<And>x. x \<in> A \<Longrightarrow> f x = g x"
c6348a062131 tuned whitespace;
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parents: 58606
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   285
  shows "f = g"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
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   286
  using assms by (force simp add: fun_eq_iff extensional_def)
14853
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parents: 14762
diff changeset
   287
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9f86e46779e4 new lemmas
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   288
lemma extensional_restrict:  "f \<in> extensional A \<Longrightarrow> restrict f A = f"
58783
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   289
  by (rule extensionalityI[OF restrict_extensional]) auto
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parents: 39302
diff changeset
   290
50123
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   291
lemma extensional_subset: "f \<in> extensional A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f \<in> extensional B"
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   292
  unfolding extensional_def by auto
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
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   293
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lemma inv_into_funcset: "f ` A = B \<Longrightarrow> (\<lambda>x\<in>B. inv_into A f x) \<in> B \<rightarrow> A"
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parents: 58606
diff changeset
   295
  by (unfold inv_into_def) (fast intro: someI2)
14853
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paulson
parents: 14762
diff changeset
   296
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parents: 58606
diff changeset
   297
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)"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   298
  apply (simp add: bij_betw_def compose_def)
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   299
  apply (rule restrict_ext, auto)
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wenzelm
parents: 58606
diff changeset
   300
  done
14853
8d710bece29f more on bij_betw
paulson
parents: 14762
diff changeset
   301
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parents: 58606
diff changeset
   302
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)"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   303
  apply (simp add: compose_def)
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wenzelm
parents: 58606
diff changeset
   304
  apply (rule restrict_ext)
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   305
  apply (simp add: f_inv_into_f)
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   306
  done
14853
8d710bece29f more on bij_betw
paulson
parents: 14762
diff changeset
   307
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   308
lemma extensional_insert[intro, simp]:
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   309
  assumes "a \<in> extensional (insert i I)"
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   310
  shows "a(i := b) \<in> extensional (insert i I)"
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   311
  using assms unfolding extensional_def by auto
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   312
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parents: 58606
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   313
lemma extensional_Int[simp]: "extensional I \<inter> extensional I' = extensional (I \<inter> I')"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   314
  unfolding extensional_def by auto
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   315
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   316
lemma extensional_UNIV[simp]: "extensional UNIV = UNIV"
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   317
  by (auto simp: extensional_def)
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   318
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   319
lemma restrict_extensional_sub[intro]: "A \<subseteq> B \<Longrightarrow> restrict f A \<in> extensional B"
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   320
  unfolding restrict_def extensional_def by auto
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   321
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   322
lemma extensional_insert_undefined[intro, simp]:
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   323
  "a \<in> extensional (insert i I) \<Longrightarrow> a(i := undefined) \<in> extensional I"
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   324
  unfolding extensional_def by auto
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   325
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   326
lemma extensional_insert_cancel[intro, simp]:
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   327
  "a \<in> extensional I \<Longrightarrow> a \<in> extensional (insert i I)"
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   328
  unfolding extensional_def by auto
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   329
14762
bd349ff7907a new bij_betw operator
paulson
parents: 14745
diff changeset
   330
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   331
subsection \<open>Cardinality\<close>
14745
94be403deb84 new lemmas
paulson
parents: 14706
diff changeset
   332
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   333
lemma card_inj: "f \<in> A \<rightarrow> B \<Longrightarrow> inj_on f A \<Longrightarrow> finite B \<Longrightarrow> card A \<le> card B"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   334
  by (rule card_inj_on_le) auto
14745
94be403deb84 new lemmas
paulson
parents: 14706
diff changeset
   335
94be403deb84 new lemmas
paulson
parents: 14706
diff changeset
   336
lemma card_bij:
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   337
  assumes "f \<in> A \<rightarrow> B" "inj_on f A"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   338
    and "g \<in> B \<rightarrow> A" "inj_on g B"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   339
    and "finite A" "finite B"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   340
  shows "card A = card B"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   341
  using assms by (blast intro: card_inj order_antisym)
14745
94be403deb84 new lemmas
paulson
parents: 14706
diff changeset
   342
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   343
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   344
subsection \<open>Extensional Function Spaces\<close>
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   345
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   346
definition PiE :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow> 'b) set"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   347
  where "PiE S T = Pi S T \<inter> extensional S"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   348
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 50123
diff changeset
   349
abbreviation "Pi\<^sub>E A B \<equiv> PiE A B"
40631
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bulwahn
parents: 39595
diff changeset
   350
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61585
diff changeset
   351
syntax
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   352
  "_PiE" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(3\<Pi>\<^sub>E _\<in>_./ _)" 10)
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61585
diff changeset
   353
translations
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61585
diff changeset
   354
  "\<Pi>\<^sub>E x\<in>A. B" \<rightleftharpoons> "CONST Pi\<^sub>E A (\<lambda>x. B)"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   355
61384
9f5145281888 prefer symbols;
wenzelm
parents: 61378
diff changeset
   356
abbreviation extensional_funcset :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<Rightarrow> 'b) set" (infixr "\<rightarrow>\<^sub>E" 60)
9f5145281888 prefer symbols;
wenzelm
parents: 61378
diff changeset
   357
  where "A \<rightarrow>\<^sub>E B \<equiv> (\<Pi>\<^sub>E i\<in>A. B)"
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   358
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   359
lemma extensional_funcset_def: "extensional_funcset S T = (S \<rightarrow> T) \<inter> extensional S"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   360
  by (simp add: PiE_def)
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   361
64910
6108dddad9f0 more symbols via abbrevs;
wenzelm
parents: 63092
diff changeset
   362
lemma PiE_empty_domain[simp]: "Pi\<^sub>E {} T = {\<lambda>x. undefined}"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   363
  unfolding PiE_def by simp
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   364
64910
6108dddad9f0 more symbols via abbrevs;
wenzelm
parents: 63092
diff changeset
   365
lemma PiE_UNIV_domain: "Pi\<^sub>E UNIV T = Pi UNIV T"
54417
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 53381
diff changeset
   366
  unfolding PiE_def by simp
dbb8ecfe1337 add restrict_space measure
hoelzl
parents: 53381
diff changeset
   367
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   368
lemma PiE_empty_range[simp]: "i \<in> I \<Longrightarrow> F i = {} \<Longrightarrow> (\<Pi>\<^sub>E i\<in>I. F i) = {}"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   369
  unfolding PiE_def by auto
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   370
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   371
lemma PiE_eq_empty_iff: "Pi\<^sub>E I F = {} \<longleftrightarrow> (\<exists>i\<in>I. F i = {})"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   372
proof
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 50123
diff changeset
   373
  assume "Pi\<^sub>E I F = {}"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   374
  show "\<exists>i\<in>I. F i = {}"
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   375
  proof (rule ccontr)
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   376
    assume "\<not> ?thesis"
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   377
    then have "\<forall>i. \<exists>y. (i \<in> I \<longrightarrow> y \<in> F i) \<and> (i \<notin> I \<longrightarrow> y = undefined)"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   378
      by auto
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53015
diff changeset
   379
    from choice[OF this]
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53015
diff changeset
   380
    obtain f where " \<forall>x. (x \<in> I \<longrightarrow> f x \<in> F x) \<and> (x \<notin> I \<longrightarrow> f x = undefined)" ..
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   381
    then have "f \<in> Pi\<^sub>E I F"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   382
      by (auto simp: extensional_def PiE_def)
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   383
    with \<open>Pi\<^sub>E I F = {}\<close> show False
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   384
      by auto
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   385
  qed
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   386
qed (auto simp: PiE_def)
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   387
64910
6108dddad9f0 more symbols via abbrevs;
wenzelm
parents: 63092
diff changeset
   388
lemma PiE_arb: "f \<in> Pi\<^sub>E S T \<Longrightarrow> x \<notin> S \<Longrightarrow> f x = undefined"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   389
  unfolding PiE_def by auto (auto dest!: extensional_arb)
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   390
64910
6108dddad9f0 more symbols via abbrevs;
wenzelm
parents: 63092
diff changeset
   391
lemma PiE_mem: "f \<in> Pi\<^sub>E S T \<Longrightarrow> x \<in> S \<Longrightarrow> f x \<in> T x"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   392
  unfolding PiE_def by auto
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   393
64910
6108dddad9f0 more symbols via abbrevs;
wenzelm
parents: 63092
diff changeset
   394
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"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   395
  unfolding PiE_def extensional_def by auto
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   396
64910
6108dddad9f0 more symbols via abbrevs;
wenzelm
parents: 63092
diff changeset
   397
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"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   398
  unfolding PiE_def extensional_def by auto
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   399
64910
6108dddad9f0 more symbols via abbrevs;
wenzelm
parents: 63092
diff changeset
   400
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)"
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   401
proof -
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   402
  {
64910
6108dddad9f0 more symbols via abbrevs;
wenzelm
parents: 63092
diff changeset
   403
    fix f assume "f \<in> Pi\<^sub>E (insert x S) T" "x \<notin> S"
6108dddad9f0 more symbols via abbrevs;
wenzelm
parents: 63092
diff changeset
   404
    then have "f \<in> (\<lambda>(y, g). g(x := y)) ` (T x \<times> Pi\<^sub>E S T)"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   405
      by (auto intro!: image_eqI[where x="(f x, f(x := undefined))"] intro: fun_upd_in_PiE PiE_mem)
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   406
  }
59425
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 58881
diff changeset
   407
  moreover
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 58881
diff changeset
   408
  {
64910
6108dddad9f0 more symbols via abbrevs;
wenzelm
parents: 63092
diff changeset
   409
    fix f assume "f \<in> Pi\<^sub>E (insert x S) T" "x \<in> S"
6108dddad9f0 more symbols via abbrevs;
wenzelm
parents: 63092
diff changeset
   410
    then have "f \<in> (\<lambda>(y, g). g(x := y)) ` (T x \<times> Pi\<^sub>E S T)"
59425
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 58881
diff changeset
   411
      by (auto intro!: image_eqI[where x="(f x, f)"] intro: fun_upd_in_PiE PiE_mem simp: insert_absorb)
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 58881
diff changeset
   412
  }
c5e79df8cc21 import general thms from Density_Compiler
hoelzl
parents: 58881
diff changeset
   413
  ultimately show ?thesis
63092
a949b2a5f51d eliminated use of empty "assms";
wenzelm
parents: 63060
diff changeset
   414
    by (auto intro: PiE_fun_upd)
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   415
qed
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   416
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   417
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)"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   418
  by (auto simp: PiE_def)
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   419
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   420
lemma PiE_cong: "(\<And>i. i\<in>I \<Longrightarrow> A i = B i) \<Longrightarrow> Pi\<^sub>E I A = Pi\<^sub>E I B"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   421
  unfolding PiE_def by (auto simp: Pi_cong)
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   422
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   423
lemma PiE_E [elim]:
64910
6108dddad9f0 more symbols via abbrevs;
wenzelm
parents: 63092
diff changeset
   424
  assumes "f \<in> Pi\<^sub>E A B"
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   425
  obtains "x \<in> A" and "f x \<in> B x"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   426
    | "x \<notin> A" and "f x = undefined"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   427
  using assms by (auto simp: Pi_def PiE_def extensional_def)
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   428
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   429
lemma PiE_I[intro!]:
64910
6108dddad9f0 more symbols via abbrevs;
wenzelm
parents: 63092
diff changeset
   430
  "(\<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"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   431
  by (simp add: PiE_def extensional_def)
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   432
64910
6108dddad9f0 more symbols via abbrevs;
wenzelm
parents: 63092
diff changeset
   433
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"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   434
  by auto
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   435
64910
6108dddad9f0 more symbols via abbrevs;
wenzelm
parents: 63092
diff changeset
   436
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"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   437
  by (simp add: PiE_def Pi_iff)
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   438
73348
65c45cba3f54 reverted simprule status on a new lemma
paulson <lp15@cam.ac.uk>
parents: 73346
diff changeset
   439
lemma restrict_PiE_iff: "restrict f I \<in> Pi\<^sub>E I X \<longleftrightarrow> (\<forall>i \<in> I. f i \<in> X i)"
73346
00e0f7724c06 tiny bit of lemma hacking
paulson <lp15@cam.ac.uk>
parents: 71838
diff changeset
   440
  by (simp add: PiE_iff)
00e0f7724c06 tiny bit of lemma hacking
paulson <lp15@cam.ac.uk>
parents: 71838
diff changeset
   441
71258
d67924987c34 a few new and tidier proofs (mostly about finite sets)
paulson <lp15@cam.ac.uk>
parents: 70063
diff changeset
   442
lemma ext_funcset_to_sing_iff [simp]: "A \<rightarrow>\<^sub>E {a} = {\<lambda>x\<in>A. a}"
d67924987c34 a few new and tidier proofs (mostly about finite sets)
paulson <lp15@cam.ac.uk>
parents: 70063
diff changeset
   443
  by (auto simp: PiE_def Pi_iff extensionalityI)
d67924987c34 a few new and tidier proofs (mostly about finite sets)
paulson <lp15@cam.ac.uk>
parents: 70063
diff changeset
   444
64910
6108dddad9f0 more symbols via abbrevs;
wenzelm
parents: 63092
diff changeset
   445
lemma PiE_restrict[simp]:  "f \<in> Pi\<^sub>E A B \<Longrightarrow> restrict f A = f"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   446
  by (simp add: extensional_restrict PiE_def)
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   447
64910
6108dddad9f0 more symbols via abbrevs;
wenzelm
parents: 63092
diff changeset
   448
lemma restrict_PiE[simp]: "restrict f I \<in> Pi\<^sub>E I S \<longleftrightarrow> f \<in> Pi I S"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   449
  by (auto simp: PiE_iff)
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   450
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   451
lemma PiE_eq_subset:
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   452
  assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   453
    and eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   454
    and "i \<in> I"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   455
  shows "F i \<subseteq> F' i"
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   456
proof
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   457
  fix x
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   458
  assume "x \<in> F i"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   459
  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)"
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53015
diff changeset
   460
    by auto
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53015
diff changeset
   461
  from choice[OF this] obtain f
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53015
diff changeset
   462
    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)" ..
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   463
  then have "f \<in> Pi\<^sub>E I F"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   464
    by (auto simp: extensional_def PiE_def)
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   465
  then have "f \<in> Pi\<^sub>E I F'"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   466
    using assms by simp
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   467
  then show "x \<in> F' i"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   468
    using f \<open>i \<in> I\<close> by (auto simp: PiE_def)
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   469
qed
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   470
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   471
lemma PiE_eq_iff_not_empty:
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   472
  assumes ne: "\<And>i. i \<in> I \<Longrightarrow> F i \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> F' i \<noteq> {}"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 50123
diff changeset
   473
  shows "Pi\<^sub>E I F = Pi\<^sub>E I F' \<longleftrightarrow> (\<forall>i\<in>I. F i = F' i)"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   474
proof (intro iffI ballI)
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   475
  fix i
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   476
  assume eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   477
  assume i: "i \<in> I"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   478
  show "F i = F' i"
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   479
    using PiE_eq_subset[of I F F', OF ne eq i]
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   480
    using PiE_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   481
    by auto
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   482
qed (auto simp: PiE_def)
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   483
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   484
lemma PiE_eq_iff:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 50123
diff changeset
   485
  "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 = {}))"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   486
proof (intro iffI disjCI)
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 50123
diff changeset
   487
  assume eq[simp]: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   488
  assume "\<not> ((\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   489
  then have "(\<forall>i\<in>I. F i \<noteq> {}) \<and> (\<forall>i\<in>I. F' i \<noteq> {})"
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   490
    using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by auto
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   491
  with PiE_eq_iff_not_empty[of I F F'] show "\<forall>i\<in>I. F i = F' i"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   492
    by auto
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   493
next
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   494
  assume "(\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {})"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 50123
diff changeset
   495
  then show "Pi\<^sub>E I F = Pi\<^sub>E I F'"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   496
    using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by (auto simp: PiE_def)
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   497
qed
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   498
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   499
lemma extensional_funcset_fun_upd_restricts_rangeI:
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   500
  "\<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})"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   501
  unfolding extensional_funcset_def extensional_def
73346
00e0f7724c06 tiny bit of lemma hacking
paulson <lp15@cam.ac.uk>
parents: 71838
diff changeset
   502
  by (auto split: if_split_asm)
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   503
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   504
lemma extensional_funcset_fun_upd_extends_rangeI:
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 50123
diff changeset
   505
  assumes "a \<in> T" "f \<in> S \<rightarrow>\<^sub>E (T - {a})"
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   506
  shows "f(x := a) \<in> insert x S \<rightarrow>\<^sub>E  T"
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   507
  using assms unfolding extensional_funcset_def extensional_def by auto
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   508
69000
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   509
lemma subset_PiE:
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   510
   "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")
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   511
proof (cases "PiE I S = {}")
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   512
  case False
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   513
  moreover have "?lhs = ?rhs"
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   514
  proof
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   515
    assume L: ?lhs
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   516
    have "\<And>i. i\<in>I \<Longrightarrow> S i \<noteq> {}"
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   517
      using False PiE_eq_empty_iff by blast
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   518
    with L show ?rhs
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   519
      by (simp add: PiE_Int PiE_eq_iff inf.absorb_iff2)
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   520
  qed auto
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   521
  ultimately show ?thesis
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   522
    by simp
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   523
qed simp
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   524
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   525
lemma PiE_eq:
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   526
   "PiE I S = PiE I T \<longleftrightarrow> PiE I S = {} \<and> PiE I T = {} \<or> (\<forall>i \<in> I. S i = T i)"
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   527
  by (auto simp: PiE_eq_iff PiE_eq_empty_iff)
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   528
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   529
lemma PiE_UNIV [simp]: "PiE UNIV (\<lambda>i. UNIV) = UNIV"
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   530
  by blast
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   531
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   532
lemma image_projection_PiE:
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   533
  "(\<lambda>f. f i) ` (PiE I S) = (if PiE I S = {} then {} else if i \<in> I then S i else {undefined})"
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   534
proof -
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   535
  have "(\<lambda>f. f i) ` Pi\<^sub>E I S = S i" if "i \<in> I" "f \<in> PiE I S" for f
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   536
    using that apply auto
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   537
    by (rule_tac x="(\<lambda>k. if k=i then x else f k)" in image_eqI) auto
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   538
  moreover have "(\<lambda>f. f i) ` Pi\<^sub>E I S = {undefined}" if "f \<in> PiE I S" "i \<notin> I" for f
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   539
    using that by (blast intro: PiE_arb [OF that, symmetric])
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   540
  ultimately show ?thesis
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   541
    by auto
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   542
qed
7cb3ddd60fd6 more lemmas
paulson <lp15@cam.ac.uk>
parents: 68687
diff changeset
   543
73348
65c45cba3f54 reverted simprule status on a new lemma
paulson <lp15@cam.ac.uk>
parents: 73346
diff changeset
   544
lemma PiE_singleton:
69710
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   545
  assumes "f \<in> extensional A"
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   546
  shows   "PiE A (\<lambda>x. {f x}) = {f}"
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   547
proof -
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   548
  {
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   549
    fix g assume "g \<in> PiE A (\<lambda>x. {f x})"
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   550
    hence "g x = f x" for x
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   551
      using assms by (cases "x \<in> A") (auto simp: extensional_def)
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   552
    hence "g = f" by (simp add: fun_eq_iff)
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   553
  }
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   554
  thus ?thesis using assms by (auto simp: extensional_def)
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   555
qed
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   556
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   557
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})"
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   558
  by (metis (mono_tags, lifting) PiE_eq PiE_singleton insert_not_empty restrict_apply' restrict_extensional)
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   559
69939
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69710
diff changeset
   560
lemma PiE_over_singleton_iff: "(\<Pi>\<^sub>E x\<in>{a}. B x) = (\<Union>b \<in> B a. {\<lambda>x \<in> {a}. b})"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69710
diff changeset
   561
  apply (auto simp: PiE_iff split: if_split_asm)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69710
diff changeset
   562
  apply (metis (no_types, lifting) extensionalityI restrict_apply' restrict_extensional singletonD)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69710
diff changeset
   563
  done
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69710
diff changeset
   564
69710
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   565
lemma all_PiE_elements:
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   566
   "(\<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")
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   567
proof (cases "PiE I S = {}")
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   568
  case False
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   569
  then obtain f where f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> S i"
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   570
    by fastforce
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   571
  show ?thesis
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   572
  proof
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   573
    assume L: ?lhs
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   574
    have "P i x"
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   575
      if "i \<in> I" "x \<in> S i" for i x
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   576
    proof -
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   577
      have "(\<lambda>j \<in> I. if j=i then x else f j) \<in> PiE I S"
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   578
        by (simp add: f that(2))
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   579
      then have "P i ((\<lambda>j \<in> I. if j=i then x else f j) i)"
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   580
        using L that(1) by blast
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   581
      with that show ?thesis
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   582
        by simp
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   583
    qed
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   584
    then show ?rhs
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   585
      by (simp add: False)
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   586
  qed fastforce
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   587
qed simp
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   588
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   589
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"
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   590
  by (metis ext PiE_E)
61372780515b some renamings and a bit of new material
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   591
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   592
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   593
subsubsection \<open>Injective Extensional Function Spaces\<close>
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   594
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   595
lemma extensional_funcset_fun_upd_inj_onI:
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   596
  assumes "f \<in> S \<rightarrow>\<^sub>E (T - {a})"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   597
    and "inj_on f S"
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   598
  shows "inj_on (f(x := a)) S"
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   599
  using assms
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   600
  unfolding extensional_funcset_def by (auto intro!: inj_on_fun_updI)
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   601
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   602
lemma extensional_funcset_extend_domain_inj_on_eq:
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   603
  assumes "x \<notin> S"
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   604
  shows "{f. f \<in> (insert x S) \<rightarrow>\<^sub>E T \<and> inj_on f (insert x S)} =
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   605
    (\<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}"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   606
  using assms
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   607
  apply (auto del: PiE_I PiE_E)
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   608
  apply (auto intro: extensional_funcset_fun_upd_inj_onI
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   609
    extensional_funcset_fun_upd_extends_rangeI del: PiE_I PiE_E)
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   610
  apply (auto simp add: image_iff inj_on_def)
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   611
  apply (rule_tac x="xa x" in exI)
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   612
  apply (auto intro: PiE_mem del: PiE_I PiE_E)
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   613
  apply (rule_tac x="xa(x := undefined)" in exI)
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   614
  apply (auto intro!: extensional_funcset_fun_upd_restricts_rangeI)
62390
842917225d56 more canonical names
nipkow
parents: 61955
diff changeset
   615
  apply (auto dest!: PiE_mem split: if_split_asm)
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   616
  done
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   617
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   618
lemma extensional_funcset_extend_domain_inj_onI:
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   619
  assumes "x \<notin> S"
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 50123
diff changeset
   620
  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}"
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   621
  using assms
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   622
  apply (auto intro!: inj_onI)
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   623
  apply (metis fun_upd_same)
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   624
  apply (metis assms PiE_arb fun_upd_triv fun_upd_upd)
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   625
  done
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   626
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   627
69144
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   628
subsubsection \<open>Misc properties of functions, composition and restriction from HOL Light\<close>
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   629
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   630
lemma function_factors_left_gen:
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   631
  "(\<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))"
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   632
  (is "?lhs = ?rhs")
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   633
proof
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   634
  assume L: ?lhs
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   635
  then show ?rhs
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   636
    apply (rule_tac x="f \<circ> inv_into (Collect P) g" in exI)
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   637
    unfolding o_def
73932
fd21b4a93043 added opaque_combs and renamed hide_lams to opaque_lifting
desharna
parents: 73348
diff changeset
   638
    by (metis (mono_tags, opaque_lifting) f_inv_into_f imageI inv_into_into mem_Collect_eq)
69144
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   639
qed auto
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   640
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   641
lemma function_factors_left:
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   642
  "(\<forall>x y. (g x = g y) \<longrightarrow> (f x = f y)) \<longleftrightarrow> (\<exists>h. f = h \<circ> g)"
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   643
  using function_factors_left_gen [of "\<lambda>x. True" g f] unfolding o_def by blast
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   644
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   645
lemma function_factors_right_gen:
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   646
  "(\<forall>x. P x \<longrightarrow> (\<exists>y. g y = f x)) \<longleftrightarrow> (\<exists>h. \<forall>x. P x \<longrightarrow> f x = g(h x))"
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   647
  by metis
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   648
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   649
lemma function_factors_right:
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   650
  "(\<forall>x. \<exists>y. g y = f x) \<longleftrightarrow> (\<exists>h. f = g \<circ> h)"
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   651
  unfolding o_def by metis
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   652
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   653
lemma restrict_compose_right:
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   654
   "restrict (g \<circ> restrict f S) S = restrict (g \<circ> f) S"
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   655
  by auto
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   656
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   657
lemma restrict_compose_left:
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   658
   "f ` S \<subseteq> T \<Longrightarrow> restrict (restrict g T \<circ> f) S = restrict (g \<circ> f) S"
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   659
  by fastforce
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   660
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69000
diff changeset
   661
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   662
subsubsection \<open>Cardinality\<close>
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   663
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   664
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)"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   665
  by (induct S arbitrary: T rule: finite_induct) (simp_all add: PiE_insert_eq)
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   666
53015
a1119cf551e8 standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents: 50123
diff changeset
   667
lemma inj_combinator: "x \<notin> S \<Longrightarrow> inj_on (\<lambda>(y, g). g(x := y)) (T x \<times> Pi\<^sub>E S T)"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   668
proof (safe intro!: inj_onI ext)
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   669
  fix f y g z
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   670
  assume "x \<notin> S"
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   671
  assume fg: "f \<in> Pi\<^sub>E S T" "g \<in> Pi\<^sub>E S T"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   672
  assume "f(x := y) = g(x := z)"
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   673
  then have *: "\<And>i. (f(x := y)) i = (g(x := z)) i"
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   674
    unfolding fun_eq_iff by auto
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   675
  from this[of x] show "y = z" by simp
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   676
  fix i from *[of i] \<open>x \<notin> S\<close> fg show "f i = g i"
62390
842917225d56 more canonical names
nipkow
parents: 61955
diff changeset
   677
    by (auto split: if_split_asm simp: PiE_def extensional_def)
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   678
qed
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   679
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   680
lemma card_PiE: "finite S \<Longrightarrow> card (\<Pi>\<^sub>E i \<in> S. T i) = (\<Prod> i\<in>S. card (T i))"
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   681
proof (induct rule: finite_induct)
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   682
  case empty
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   683
  then show ?case by auto
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   684
next
58783
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   685
  case (insert x S)
c6348a062131 tuned whitespace;
wenzelm
parents: 58606
diff changeset
   686
  then show ?case
50123
69b35a75caf3 merge extensional dependent function space from FuncSet with the one in Finite_Product_Measure
hoelzl
parents: 50104
diff changeset
   687
    by (simp add: PiE_insert_eq inj_combinator card_image card_cartesian_product)
40631
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   688
qed
b3f85ba3dae4 adding extensional function spaces to the FuncSet library theory
bulwahn
parents: 39595
diff changeset
   689
75663
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   690
lemma card_funcsetE: "finite A \<Longrightarrow> card (A \<rightarrow>\<^sub>E B) = card B ^ card A" 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   691
  by (subst card_PiE, auto)
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   692
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   693
lemma card_inj_on_subset_funcset: assumes finB: "finite B"
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   694
  and finC: "finite C" 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   695
  and AB: "A \<subseteq> B" 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   696
shows "card {f \<in> B \<rightarrow>\<^sub>E C. inj_on f A} = 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   697
  card C^(card B - card A) * prod ((-) (card C)) {0 ..< card A}"
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   698
proof -
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   699
  define D where "D = B - A" 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   700
  from AB have B: "B = A \<union> D" and disj: "A \<inter> D = {}" unfolding D_def by auto
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   701
  have sub: "card B - card A = card D" unfolding D_def using finB AB
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   702
    by (metis card_Diff_subset finite_subset)
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   703
  have "finite A" "finite D" using finB unfolding B by auto
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   704
  thus ?thesis unfolding sub unfolding B using disj
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   705
  proof (induct A rule: finite_induct)
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   706
    case empty
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   707
    from card_funcsetE[OF this(1), of C] show ?case by auto
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   708
  next
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   709
    case (insert a A)
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   710
    have "{f. f \<in> insert a A \<union> D \<rightarrow>\<^sub>E C \<and> inj_on f (insert a A)}
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   711
      = {f(a := c) | f c. f \<in> A \<union> D \<rightarrow>\<^sub>E C \<and> inj_on f A \<and> c \<in> C - f ` A}" 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   712
      (is "?l = ?r")
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   713
    proof
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   714
      show "?r \<subseteq> ?l" 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   715
        by (auto intro: inj_on_fun_updI split: if_splits) 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   716
      {
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   717
        fix f
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   718
        assume f: "f \<in> ?l" 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   719
        let ?g = "f(a := undefined)" 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   720
        let ?h = "?g(a := f a)" 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   721
        have mem: "f a \<in> C - ?g ` A" using insert(1,2,4,5) f by auto
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   722
        from f have f: "f \<in> insert a A \<union> D \<rightarrow>\<^sub>E C" "inj_on f (insert a A)" by auto
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   723
        hence "?g \<in> A \<union> D \<rightarrow>\<^sub>E C" "inj_on ?g A" using \<open>a \<notin> A\<close> \<open>insert a A \<inter> D = {}\<close>
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   724
          by (auto split: if_splits simp: inj_on_def)
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   725
        with mem have "?h \<in> ?r" by blast
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   726
        also have "?h = f" by auto
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   727
        finally have "f \<in> ?r" .
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   728
      }
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   729
      thus "?l \<subseteq> ?r" by auto
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   730
    qed
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   731
    also have "\<dots> = (\<lambda> (f, c). f (a := c)) ` 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   732
         (Sigma {f . f \<in> A \<union> D \<rightarrow>\<^sub>E C \<and> inj_on f A} (\<lambda> f. C - f ` A))"
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   733
      by auto
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   734
    also have "card (...) = card (Sigma {f . f \<in> A \<union> D \<rightarrow>\<^sub>E C \<and> inj_on f A} (\<lambda> f. C - f ` A))" 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   735
    proof (rule card_image, intro inj_onI, clarsimp, goal_cases) 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   736
      case (1 f c g d)
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   737
      let ?f = "f(a := c, a := undefined)" 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   738
      let ?g = "g(a := d, a := undefined)" 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   739
      from 1 have id: "f(a := c) = g(a := d)" by auto
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   740
      from fun_upd_eqD[OF id] 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   741
      have cd: "c = d" by auto
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   742
      from id have "?f = ?g" by auto
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   743
      also have "?f = f" using `f \<in> A \<union> D \<rightarrow>\<^sub>E C` insert(1,2,4,5) 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   744
        by (intro ext, auto)
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   745
      also have "?g = g" using `g \<in> A \<union> D \<rightarrow>\<^sub>E C` insert(1,2,4,5) 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   746
        by (intro ext, auto)
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   747
      finally show "f = g \<and> c = d" using cd by auto
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   748
    qed
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   749
    also have "\<dots> = (\<Sum>f\<in>{f \<in> A \<union> D \<rightarrow>\<^sub>E C. inj_on f A}. card (C - f ` A))" 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   750
      by (rule card_SigmaI, rule finite_subset[of _ "A \<union> D \<rightarrow>\<^sub>E C"],
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   751
          insert \<open>finite C\<close> \<open>finite D\<close> \<open>finite A\<close>, auto intro!: finite_PiE)
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   752
    also have "\<dots> = (\<Sum>f\<in>{f \<in> A \<union> D \<rightarrow>\<^sub>E C. inj_on f A}. card C - card A)"
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   753
      by (rule sum.cong[OF refl], subst card_Diff_subset, insert \<open>finite A\<close>, auto simp: card_image)
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   754
    also have "\<dots> = (card C - card A) * card {f \<in> A \<union> D \<rightarrow>\<^sub>E C. inj_on f A}" 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   755
      by simp
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   756
    also have "\<dots> = card C ^ card D * ((card C - card A) * prod ((-) (card C)) {0..<card A})" 
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   757
      using insert by (auto simp: ac_simps)
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   758
    also have "(card C - card A) * prod ((-) (card C)) {0..<card A} =
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   759
      prod ((-) (card C)) {0..<Suc (card A)}" by simp
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   760
    also have "Suc (card A) = card (insert a A)" using insert by auto
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   761
    finally show ?case .
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   762
  qed
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   763
qed
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   764
f2e402a19530 moved lemmas from AFP
nipkow
parents: 75078
diff changeset
   765
71838
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   766
subsection \<open>The pigeonhole principle\<close>
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   767
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   768
text \<open>
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   769
  An alternative formulation of this is that for a function mapping a finite set \<open>A\<close> of
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   770
  cardinality \<open>m\<close> to a finite set \<open>B\<close> of cardinality \<open>n\<close>, there exists an element \<open>y \<in> B\<close> that
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   771
  is hit at least $\lceil \frac{m}{n}\rceil$ times. However, since we do not have real numbers
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   772
  or rounding yet, we state it in the following equivalent form:
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   773
\<close>
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   774
lemma pigeonhole_card:
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   775
  assumes "f \<in> A \<rightarrow> B" "finite A" "finite B" "B \<noteq> {}"
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   776
  shows   "\<exists>y\<in>B. card (f -` {y} \<inter> A) * card B \<ge> card A"
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   777
proof -
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   778
  from assms have "card B > 0"
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   779
    by auto
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   780
  define M where "M = Max ((\<lambda>y. card (f -` {y} \<inter> A)) ` B)"
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   781
  have "A = (\<Union>y\<in>B. f -` {y} \<inter> A)"
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   782
    using assms by auto
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   783
  also have "card \<dots> = (\<Sum>i\<in>B. card (f -` {i} \<inter> A))"
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   784
    using assms by (subst card_UN_disjoint) auto
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   785
  also have "\<dots> \<le> (\<Sum>i\<in>B. M)"
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   786
    unfolding M_def using assms by (intro sum_mono Max.coboundedI) auto
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   787
  also have "\<dots> = card B * M"
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   788
    by simp
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   789
  finally have "M * card B \<ge> card A"
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   790
    by (simp add: mult_ac)
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   791
  moreover have "M \<in> (\<lambda>y. card (f -` {y} \<inter> A)) ` B"
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   792
    unfolding M_def using assms \<open>B \<noteq> {}\<close> by (intro Max_in) auto
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   793
  ultimately show ?thesis
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   794
    by blast
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   795
qed
5656ec95493c generalised pigeonhole principle in HOL-Library.FuncSet
Manuel Eberl <eberlm@in.tum.de>
parents: 71258
diff changeset
   796
13586
0f339348df0e new theory for Pi-sets, restrict, etc.
paulson
parents:
diff changeset
   797
end