src/HOL/Library/FuncSet.thy
 author paulson Wed May 19 11:30:56 2004 +0200 (2004-05-19) changeset 14762 bd349ff7907a parent 14745 94be403deb84 child 14853 8d710bece29f permissions -rw-r--r--
new bij_betw operator
```     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 = Main:
```
```     9
```
```    10 constdefs
```
```    11   Pi :: "['a set, 'a => 'b set] => ('a => 'b) set"
```
```    12   "Pi A B == {f. \<forall>x. x \<in> A --> f x \<in> B x}"
```
```    13
```
```    14   extensional :: "'a set => ('a => 'b) set"
```
```    15   "extensional A == {f. \<forall>x. x~:A --> f x = arbitrary}"
```
```    16
```
```    17   "restrict" :: "['a => 'b, 'a set] => ('a => 'b)"
```
```    18   "restrict f A == (%x. if x \<in> A then f x else arbitrary)"
```
```    19
```
```    20 syntax
```
```    21   "@Pi"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI _:_./ _)" 10)
```
```    22   funcset :: "['a set, 'b set] => ('a => 'b) set"      (infixr "->" 60)
```
```    23   "@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3%_:_./ _)" [0,0,3] 3)
```
```    24
```
```    25 syntax (xsymbols)
```
```    26   "@Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
```
```    27   funcset :: "['a set, 'b set] => ('a => 'b) set"  (infixr "\<rightarrow>" 60)
```
```    28   "@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
```
```    29
```
```    30 syntax (HTML output)
```
```    31   "@Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
```
```    32   "@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
```
```    33
```
```    34 translations
```
```    35   "PI x:A. B" => "Pi A (%x. B)"
```
```    36   "A -> B" => "Pi A (_K B)"
```
```    37   "%x:A. f" == "restrict (%x. f) A"
```
```    38
```
```    39 constdefs
```
```    40   "compose" :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)"
```
```    41   "compose A g f == \<lambda>x\<in>A. g (f x)"
```
```    42
```
```    43 print_translation {* [("Pi", dependent_tr' ("@Pi", "funcset"))] *}
```
```    44
```
```    45
```
```    46 subsection{*Basic Properties of @{term Pi}*}
```
```    47
```
```    48 lemma Pi_I: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
```
```    49   by (simp add: Pi_def)
```
```    50
```
```    51 lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
```
```    52   by (simp add: Pi_def)
```
```    53
```
```    54 lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
```
```    55   by (simp add: Pi_def)
```
```    56
```
```    57 lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
```
```    58   by (simp add: Pi_def)
```
```    59
```
```    60 lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B"
```
```    61 by (auto simp add: Pi_def)
```
```    62
```
```    63 lemma Pi_eq_empty: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
```
```    64 apply (simp add: Pi_def, auto)
```
```    65 txt{*Converse direction requires Axiom of Choice to exhibit a function
```
```    66 picking an element from each non-empty @{term "B x"}*}
```
```    67 apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec, auto)
```
```    68 apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto)
```
```    69 done
```
```    70
```
```    71 lemma Pi_empty [simp]: "Pi {} B = UNIV"
```
```    72   by (simp add: Pi_def)
```
```    73
```
```    74 lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
```
```    75   by (simp add: Pi_def)
```
```    76
```
```    77 text{*Covariance of Pi-sets in their second argument*}
```
```    78 lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"
```
```    79   by (simp add: Pi_def, blast)
```
```    80
```
```    81 text{*Contravariance of Pi-sets in their first argument*}
```
```    82 lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"
```
```    83   by (simp add: Pi_def, blast)
```
```    84
```
```    85
```
```    86 subsection{*Composition With a Restricted Domain: @{term compose}*}
```
```    87
```
```    88 lemma funcset_compose:
```
```    89     "[| f \<in> A -> B; g \<in> B -> C |]==> compose A g f \<in> A -> C"
```
```    90   by (simp add: Pi_def compose_def restrict_def)
```
```    91
```
```    92 lemma compose_assoc:
```
```    93     "[| f \<in> A -> B; g \<in> B -> C; h \<in> C -> D |]
```
```    94       ==> compose A h (compose A g f) = compose A (compose B h g) f"
```
```    95   by (simp add: expand_fun_eq Pi_def compose_def restrict_def)
```
```    96
```
```    97 lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"
```
```    98   by (simp add: compose_def restrict_def)
```
```    99
```
```   100 lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
```
```   101   by (auto simp add: image_def compose_eq)
```
```   102
```
```   103 lemma inj_on_compose:
```
```   104     "[| f ` A = B; inj_on f A; inj_on g B |] ==> inj_on (compose A g f) A"
```
```   105   by (auto simp add: inj_on_def compose_eq)
```
```   106
```
```   107
```
```   108 subsection{*Bounded Abstraction: @{term restrict}*}
```
```   109
```
```   110 lemma restrict_in_funcset: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A -> B"
```
```   111   by (simp add: Pi_def restrict_def)
```
```   112
```
```   113 lemma restrictI: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
```
```   114   by (simp add: Pi_def restrict_def)
```
```   115
```
```   116 lemma restrict_apply [simp]:
```
```   117     "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else arbitrary)"
```
```   118   by (simp add: restrict_def)
```
```   119
```
```   120 lemma restrict_ext:
```
```   121     "(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
```
```   122   by (simp add: expand_fun_eq Pi_def Pi_def restrict_def)
```
```   123
```
```   124 lemma inj_on_restrict_eq: "inj_on (restrict f A) A = inj_on f A"
```
```   125   by (simp add: inj_on_def restrict_def)
```
```   126
```
```   127 lemma Id_compose:
```
```   128     "[|f \<in> A -> B;  f \<in> extensional A|] ==> compose A (\<lambda>y\<in>B. y) f = f"
```
```   129   by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
```
```   130
```
```   131 lemma compose_Id:
```
```   132     "[|g \<in> A -> B;  g \<in> extensional A|] ==> compose A g (\<lambda>x\<in>A. x) = g"
```
```   133   by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
```
```   134
```
```   135
```
```   136 subsection{*Extensionality*}
```
```   137
```
```   138 lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = arbitrary"
```
```   139   by (simp add: extensional_def)
```
```   140
```
```   141 lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
```
```   142   by (simp add: restrict_def extensional_def)
```
```   143
```
```   144 lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
```
```   145   by (simp add: compose_def)
```
```   146
```
```   147 lemma extensionalityI:
```
```   148     "[| f \<in> extensional A; g \<in> extensional A;
```
```   149       !!x. x\<in>A ==> f x = g x |] ==> f = g"
```
```   150   by (force simp add: expand_fun_eq extensional_def)
```
```   151
```
```   152 lemma Inv_funcset: "f ` A = B ==> (\<lambda>x\<in>B. Inv A f x) : B -> A"
```
```   153   by (unfold Inv_def) (fast intro: restrict_in_funcset someI2)
```
```   154
```
```   155 lemma compose_Inv_id:
```
```   156     "[| inj_on f A;  f ` A = B |]
```
```   157       ==> compose A (\<lambda>y\<in>B. Inv A f y) f = (\<lambda>x\<in>A. x)"
```
```   158   apply (simp add: compose_def)
```
```   159   apply (rule restrict_ext, auto)
```
```   160   apply (erule subst)
```
```   161   apply (simp add: Inv_f_f)
```
```   162   done
```
```   163
```
```   164 lemma compose_id_Inv:
```
```   165     "f ` A = B ==> compose B f (\<lambda>y\<in>B. Inv A f y) = (\<lambda>x\<in>B. x)"
```
```   166   apply (simp add: compose_def)
```
```   167   apply (rule restrict_ext)
```
```   168   apply (simp add: f_Inv_f)
```
```   169   done
```
```   170
```
```   171
```
```   172 subsection{*Bijections Between Sets*}
```
```   173
```
```   174 text{*The basic definition could be moved to @{text "Fun.thy"}, but most of
```
```   175 the theorems belong here, or need at least @{term Hilbert_Choice}.*}
```
```   176
```
```   177 constdefs
```
```   178   bij_betw :: "['a => 'b, 'a set, 'b set] => bool"         (*bijective*)
```
```   179     "bij_betw f A B == inj_on f A & f ` A = B"
```
```   180
```
```   181 lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
```
```   182 by (simp add: bij_betw_def)
```
```   183
```
```   184 lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
```
```   185 by (auto simp add: bij_betw_def inj_on_Inv Pi_def)
```
```   186
```
```   187 lemma bij_betw_Inv: "bij_betw f A B \<Longrightarrow> bij_betw (Inv A f) B A"
```
```   188 apply (auto simp add: bij_betw_def inj_on_Inv Inv_mem)
```
```   189 apply (simp add: image_compose [symmetric] o_def)
```
```   190 apply (simp add: image_def Inv_f_f)
```
```   191 done
```
```   192
```
```   193 lemma bij_betw_compose:
```
```   194     "[| bij_betw f A B; bij_betw g B C |] ==> bij_betw (compose A g f) A C"
```
```   195 apply (simp add: bij_betw_def compose_eq inj_on_compose)
```
```   196 apply (auto simp add: compose_def image_def)
```
```   197 done
```
```   198
```
```   199
```
```   200 subsection{*Cardinality*}
```
```   201
```
```   202 lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
```
```   203 apply (rule card_inj_on_le)
```
```   204 apply (auto simp add: Pi_def)
```
```   205 done
```
```   206
```
```   207 lemma card_bij:
```
```   208      "[|f \<in> A\<rightarrow>B; inj_on f A;
```
```   209         g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
```
```   210 by (blast intro: card_inj order_antisym)
```
```   211
```
```   212 end
```