src/HOL/Library/FuncSet.thy
author paulson
Tue Jun 28 15:27:45 2005 +0200 (2005-06-28)
changeset 16587 b34c8aa657a5
parent 15140 322485b816ac
child 17781 32bb237158a5
permissions -rw-r--r--
Constant "If" is now local
     1 (*  Title:      HOL/Library/FuncSet.thy
     2     ID:         $Id$
     3     Author:     Florian Kammueller and Lawrence C Paulson
     4 *)
     5 
     6 header {* Pi and Function Sets *}
     7 
     8 theory FuncSet
     9 imports Main
    10 begin
    11 
    12 constdefs
    13   Pi :: "['a set, 'a => 'b set] => ('a => 'b) set"
    14   "Pi A B == {f. \<forall>x. x \<in> A --> f x \<in> B x}"
    15 
    16   extensional :: "'a set => ('a => 'b) set"
    17   "extensional A == {f. \<forall>x. x~:A --> f x = arbitrary}"
    18 
    19   "restrict" :: "['a => 'b, 'a set] => ('a => 'b)"
    20   "restrict f A == (%x. if x \<in> A then f x else arbitrary)"
    21 
    22 syntax
    23   "@Pi"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI _:_./ _)" 10)
    24   funcset :: "['a set, 'b set] => ('a => 'b) set"      (infixr "->" 60)
    25   "@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3%_:_./ _)" [0,0,3] 3)
    26 
    27 syntax (xsymbols)
    28   "@Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
    29   funcset :: "['a set, 'b set] => ('a => 'b) set"  (infixr "\<rightarrow>" 60)
    30   "@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
    31 
    32 syntax (HTML output)
    33   "@Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
    34   "@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
    35 
    36 translations
    37   "PI x:A. B" => "Pi A (%x. B)"
    38   "A -> B" => "Pi A (_K B)"
    39   "%x:A. f" == "restrict (%x. f) A"
    40 
    41 constdefs
    42   "compose" :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)"
    43   "compose A g f == \<lambda>x\<in>A. g (f x)"
    44 
    45 print_translation {* [("Pi", dependent_tr' ("@Pi", "funcset"))] *}
    46 
    47 
    48 subsection{*Basic Properties of @{term Pi}*}
    49 
    50 lemma Pi_I: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
    51   by (simp add: Pi_def)
    52 
    53 lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
    54   by (simp add: Pi_def)
    55 
    56 lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
    57   by (simp add: Pi_def)
    58 
    59 lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
    60   by (simp add: Pi_def)
    61 
    62 lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B"
    63 by (auto simp add: Pi_def)
    64 
    65 lemma Pi_eq_empty: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
    66 apply (simp add: Pi_def, auto)
    67 txt{*Converse direction requires Axiom of Choice to exhibit a function
    68 picking an element from each non-empty @{term "B x"}*}
    69 apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec, auto)
    70 apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto)
    71 done
    72 
    73 lemma Pi_empty [simp]: "Pi {} B = UNIV"
    74   by (simp add: Pi_def)
    75 
    76 lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
    77   by (simp add: Pi_def)
    78 
    79 text{*Covariance of Pi-sets in their second argument*}
    80 lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"
    81   by (simp add: Pi_def, blast)
    82 
    83 text{*Contravariance of Pi-sets in their first argument*}
    84 lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"
    85   by (simp add: Pi_def, blast)
    86 
    87 
    88 subsection{*Composition With a Restricted Domain: @{term compose}*}
    89 
    90 lemma funcset_compose:
    91     "[| f \<in> A -> B; g \<in> B -> C |]==> compose A g f \<in> A -> C"
    92   by (simp add: Pi_def compose_def restrict_def)
    93 
    94 lemma compose_assoc:
    95     "[| f \<in> A -> B; g \<in> B -> C; h \<in> C -> D |]
    96       ==> compose A h (compose A g f) = compose A (compose B h g) f"
    97   by (simp add: expand_fun_eq Pi_def compose_def restrict_def)
    98 
    99 lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"
   100   by (simp add: compose_def restrict_def)
   101 
   102 lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
   103   by (auto simp add: image_def compose_eq)
   104 
   105 
   106 subsection{*Bounded Abstraction: @{term restrict}*}
   107 
   108 lemma restrict_in_funcset: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A -> B"
   109   by (simp add: Pi_def restrict_def)
   110 
   111 lemma restrictI: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
   112   by (simp add: Pi_def restrict_def)
   113 
   114 lemma restrict_apply [simp]:
   115     "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else arbitrary)"
   116   by (simp add: restrict_def)
   117 
   118 lemma restrict_ext:
   119     "(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
   120   by (simp add: expand_fun_eq Pi_def Pi_def restrict_def)
   121 
   122 lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
   123   by (simp add: inj_on_def restrict_def)
   124 
   125 lemma Id_compose:
   126     "[|f \<in> A -> B;  f \<in> extensional A|] ==> compose A (\<lambda>y\<in>B. y) f = f"
   127   by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
   128 
   129 lemma compose_Id:
   130     "[|g \<in> A -> B;  g \<in> extensional A|] ==> compose A g (\<lambda>x\<in>A. x) = g"
   131   by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
   132 
   133 lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
   134   by (auto simp add: restrict_def) 
   135 
   136 
   137 subsection{*Bijections Between Sets*}
   138 
   139 text{*The basic definition could be moved to @{text "Fun.thy"}, but most of
   140 the theorems belong here, or need at least @{term Hilbert_Choice}.*}
   141 
   142 constdefs
   143   bij_betw :: "['a => 'b, 'a set, 'b set] => bool"         (*bijective*)
   144     "bij_betw f A B == inj_on f A & f ` A = B"
   145 
   146 lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
   147 by (simp add: bij_betw_def)
   148 
   149 lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
   150 by (auto simp add: bij_betw_def inj_on_Inv Pi_def)
   151 
   152 lemma bij_betw_Inv: "bij_betw f A B \<Longrightarrow> bij_betw (Inv A f) B A"
   153 apply (auto simp add: bij_betw_def inj_on_Inv Inv_mem) 
   154 apply (simp add: image_compose [symmetric] o_def) 
   155 apply (simp add: image_def Inv_f_f) 
   156 done
   157 
   158 lemma inj_on_compose:
   159     "[| bij_betw f A B; inj_on g B |] ==> inj_on (compose A g f) A"
   160   by (auto simp add: bij_betw_def inj_on_def compose_eq)
   161 
   162 lemma bij_betw_compose:
   163     "[| bij_betw f A B; bij_betw g B C |] ==> bij_betw (compose A g f) A C"
   164 apply (simp add: bij_betw_def compose_eq inj_on_compose)
   165 apply (auto simp add: compose_def image_def)
   166 done
   167 
   168 lemma bij_betw_restrict_eq [simp]:
   169      "bij_betw (restrict f A) A B = bij_betw f A B"
   170   by (simp add: bij_betw_def)
   171 
   172 
   173 subsection{*Extensionality*}
   174 
   175 lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = arbitrary"
   176   by (simp add: extensional_def)
   177 
   178 lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
   179   by (simp add: restrict_def extensional_def)
   180 
   181 lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
   182   by (simp add: compose_def)
   183 
   184 lemma extensionalityI:
   185     "[| f \<in> extensional A; g \<in> extensional A;
   186       !!x. x\<in>A ==> f x = g x |] ==> f = g"
   187   by (force simp add: expand_fun_eq extensional_def)
   188 
   189 lemma Inv_funcset: "f ` A = B ==> (\<lambda>x\<in>B. Inv A f x) : B -> A"
   190   by (unfold Inv_def) (fast intro: restrict_in_funcset someI2)
   191 
   192 lemma compose_Inv_id:
   193     "bij_betw f A B ==> compose A (\<lambda>y\<in>B. Inv A f y) f = (\<lambda>x\<in>A. x)"
   194   apply (simp add: bij_betw_def compose_def)
   195   apply (rule restrict_ext, auto)
   196   apply (erule subst)
   197   apply (simp add: Inv_f_f)
   198   done
   199 
   200 lemma compose_id_Inv:
   201     "f ` A = B ==> compose B f (\<lambda>y\<in>B. Inv A f y) = (\<lambda>x\<in>B. x)"
   202   apply (simp add: compose_def)
   203   apply (rule restrict_ext)
   204   apply (simp add: f_Inv_f)
   205   done
   206 
   207 
   208 subsection{*Cardinality*}
   209 
   210 lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
   211 apply (rule card_inj_on_le)
   212 apply (auto simp add: Pi_def)
   213 done
   214 
   215 lemma card_bij:
   216      "[|f \<in> A\<rightarrow>B; inj_on f A;
   217         g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
   218 by (blast intro: card_inj order_antisym)
   219 
   220 end