src/HOL/Library/FuncSet.thy
 author nipkow Mon Sep 20 21:09:25 2010 +0200 (2010-09-20) changeset 39595 9f86e46779e4 parent 39302 d7728f65b353 child 40631 b3f85ba3dae4 permissions -rw-r--r--
new lemmas
```     1 (*  Title:      HOL/Library/FuncSet.thy
```
```     2     Author:     Florian Kammueller and Lawrence C Paulson
```
```     3 *)
```
```     4
```
```     5 header {* Pi and Function Sets *}
```
```     6
```
```     7 theory FuncSet
```
```     8 imports Hilbert_Choice Main
```
```     9 begin
```
```    10
```
```    11 definition
```
```    12   Pi :: "['a set, 'a => 'b set] => ('a => 'b) set" where
```
```    13   "Pi A B = {f. \<forall>x. x \<in> A --> f x \<in> B x}"
```
```    14
```
```    15 definition
```
```    16   extensional :: "'a set => ('a => 'b) set" where
```
```    17   "extensional A = {f. \<forall>x. x~:A --> f x = undefined}"
```
```    18
```
```    19 definition
```
```    20   "restrict" :: "['a => 'b, 'a set] => ('a => 'b)" where
```
```    21   "restrict f A = (%x. if x \<in> A then f x else undefined)"
```
```    22
```
```    23 abbreviation
```
```    24   funcset :: "['a set, 'b set] => ('a => 'b) set"
```
```    25     (infixr "->" 60) where
```
```    26   "A -> B == Pi A (%_. B)"
```
```    27
```
```    28 notation (xsymbols)
```
```    29   funcset  (infixr "\<rightarrow>" 60)
```
```    30
```
```    31 syntax
```
```    32   "_Pi"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI _:_./ _)" 10)
```
```    33   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3%_:_./ _)" [0,0,3] 3)
```
```    34
```
```    35 syntax (xsymbols)
```
```    36   "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
```
```    37   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
```
```    38
```
```    39 syntax (HTML output)
```
```    40   "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
```
```    41   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
```
```    42
```
```    43 translations
```
```    44   "PI x:A. B" == "CONST Pi A (%x. B)"
```
```    45   "%x:A. f" == "CONST restrict (%x. f) A"
```
```    46
```
```    47 definition
```
```    48   "compose" :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)" where
```
```    49   "compose A g f = (\<lambda>x\<in>A. g (f x))"
```
```    50
```
```    51
```
```    52 subsection{*Basic Properties of @{term Pi}*}
```
```    53
```
```    54 lemma Pi_I[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
```
```    55   by (simp add: Pi_def)
```
```    56
```
```    57 lemma Pi_I'[simp]: "(!!x. x : A --> f x : B x) ==> f : Pi A B"
```
```    58 by(simp add:Pi_def)
```
```    59
```
```    60 lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
```
```    61   by (simp add: Pi_def)
```
```    62
```
```    63 lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
```
```    64   by (simp add: Pi_def)
```
```    65
```
```    66 lemma PiE [elim]:
```
```    67   "f : Pi A B ==> (f x : B x ==> Q) ==> (x ~: A ==> Q) ==> Q"
```
```    68 by(auto simp: Pi_def)
```
```    69
```
```    70 lemma Pi_cong:
```
```    71   "(\<And> w. w \<in> A \<Longrightarrow> f w = g w) \<Longrightarrow> f \<in> Pi A B \<longleftrightarrow> g \<in> Pi A B"
```
```    72   by (auto simp: Pi_def)
```
```    73
```
```    74 lemma funcset_id [simp]: "(\<lambda>x. x) \<in> A \<rightarrow> A"
```
```    75   by (auto intro: Pi_I)
```
```    76
```
```    77 lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
```
```    78   by (simp add: Pi_def)
```
```    79
```
```    80 lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B"
```
```    81 by auto
```
```    82
```
```    83 lemma Pi_eq_empty[simp]: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
```
```    84 apply (simp add: Pi_def, auto)
```
```    85 txt{*Converse direction requires Axiom of Choice to exhibit a function
```
```    86 picking an element from each non-empty @{term "B x"}*}
```
```    87 apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec, auto)
```
```    88 apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto)
```
```    89 done
```
```    90
```
```    91 lemma Pi_empty [simp]: "Pi {} B = UNIV"
```
```    92 by (simp add: Pi_def)
```
```    93
```
```    94 lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
```
```    95 by (simp add: Pi_def)
```
```    96 (*
```
```    97 lemma funcset_id [simp]: "(%x. x): A -> A"
```
```    98   by (simp add: Pi_def)
```
```    99 *)
```
```   100 text{*Covariance of Pi-sets in their second argument*}
```
```   101 lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"
```
```   102 by auto
```
```   103
```
```   104 text{*Contravariance of Pi-sets in their first argument*}
```
```   105 lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"
```
```   106 by auto
```
```   107
```
```   108 lemma prod_final:
```
```   109   assumes 1: "fst \<circ> f \<in> Pi A B" and 2: "snd \<circ> f \<in> Pi A C"
```
```   110   shows "f \<in> (\<Pi> z \<in> A. B z \<times> C z)"
```
```   111 proof (rule Pi_I)
```
```   112   fix z
```
```   113   assume z: "z \<in> A"
```
```   114   have "f z = (fst (f z), snd (f z))"
```
```   115     by simp
```
```   116   also have "...  \<in> B z \<times> C z"
```
```   117     by (metis SigmaI PiE o_apply 1 2 z)
```
```   118   finally show "f z \<in> B z \<times> C z" .
```
```   119 qed
```
```   120
```
```   121
```
```   122 subsection{*Composition With a Restricted Domain: @{term compose}*}
```
```   123
```
```   124 lemma funcset_compose:
```
```   125   "[| f \<in> A -> B; g \<in> B -> C |]==> compose A g f \<in> A -> C"
```
```   126 by (simp add: Pi_def compose_def restrict_def)
```
```   127
```
```   128 lemma compose_assoc:
```
```   129     "[| f \<in> A -> B; g \<in> B -> C; h \<in> C -> D |]
```
```   130       ==> compose A h (compose A g f) = compose A (compose B h g) f"
```
```   131 by (simp add: fun_eq_iff Pi_def compose_def restrict_def)
```
```   132
```
```   133 lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"
```
```   134 by (simp add: compose_def restrict_def)
```
```   135
```
```   136 lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
```
```   137   by (auto simp add: image_def compose_eq)
```
```   138
```
```   139
```
```   140 subsection{*Bounded Abstraction: @{term restrict}*}
```
```   141
```
```   142 lemma restrict_in_funcset: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A -> B"
```
```   143   by (simp add: Pi_def restrict_def)
```
```   144
```
```   145 lemma restrictI[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
```
```   146   by (simp add: Pi_def restrict_def)
```
```   147
```
```   148 lemma restrict_apply [simp]:
```
```   149     "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else undefined)"
```
```   150   by (simp add: restrict_def)
```
```   151
```
```   152 lemma restrict_ext:
```
```   153     "(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
```
```   154   by (simp add: fun_eq_iff Pi_def restrict_def)
```
```   155
```
```   156 lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
```
```   157   by (simp add: inj_on_def restrict_def)
```
```   158
```
```   159 lemma Id_compose:
```
```   160     "[|f \<in> A -> B;  f \<in> extensional A|] ==> compose A (\<lambda>y\<in>B. y) f = f"
```
```   161   by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
```
```   162
```
```   163 lemma compose_Id:
```
```   164     "[|g \<in> A -> B;  g \<in> extensional A|] ==> compose A g (\<lambda>x\<in>A. x) = g"
```
```   165   by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
```
```   166
```
```   167 lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
```
```   168   by (auto simp add: restrict_def)
```
```   169
```
```   170
```
```   171 subsection{*Bijections Between Sets*}
```
```   172
```
```   173 text{*The definition of @{const bij_betw} is in @{text "Fun.thy"}, but most of
```
```   174 the theorems belong here, or need at least @{term Hilbert_Choice}.*}
```
```   175
```
```   176 lemma bij_betwI:
```
```   177 assumes "f \<in> A \<rightarrow> B" and "g \<in> B \<rightarrow> A"
```
```   178     and g_f: "\<And>x. x\<in>A \<Longrightarrow> g (f x) = x" and f_g: "\<And>y. y\<in>B \<Longrightarrow> f (g y) = y"
```
```   179 shows "bij_betw f A B"
```
```   180 unfolding bij_betw_def
```
```   181 proof
```
```   182   show "inj_on f A" by (metis g_f inj_on_def)
```
```   183 next
```
```   184   have "f ` A \<subseteq> B" using `f \<in> A \<rightarrow> B` by auto
```
```   185   moreover
```
```   186   have "B \<subseteq> f ` A" by auto (metis Pi_mem `g \<in> B \<rightarrow> A` f_g image_iff)
```
```   187   ultimately show "f ` A = B" by blast
```
```   188 qed
```
```   189
```
```   190 lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
```
```   191 by (auto simp add: bij_betw_def)
```
```   192
```
```   193 lemma inj_on_compose:
```
```   194   "[| bij_betw f A B; inj_on g B |] ==> inj_on (compose A g f) A"
```
```   195 by (auto simp add: bij_betw_def inj_on_def compose_eq)
```
```   196
```
```   197 lemma bij_betw_compose:
```
```   198   "[| bij_betw f A B; bij_betw g B C |] ==> bij_betw (compose A g f) A C"
```
```   199 apply (simp add: bij_betw_def compose_eq inj_on_compose)
```
```   200 apply (auto simp add: compose_def image_def)
```
```   201 done
```
```   202
```
```   203 lemma bij_betw_restrict_eq [simp]:
```
```   204   "bij_betw (restrict f A) A B = bij_betw f A B"
```
```   205 by (simp add: bij_betw_def)
```
```   206
```
```   207
```
```   208 subsection{*Extensionality*}
```
```   209
```
```   210 lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = undefined"
```
```   211 by (simp add: extensional_def)
```
```   212
```
```   213 lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
```
```   214 by (simp add: restrict_def extensional_def)
```
```   215
```
```   216 lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
```
```   217 by (simp add: compose_def)
```
```   218
```
```   219 lemma extensionalityI:
```
```   220   "[| f \<in> extensional A; g \<in> extensional A;
```
```   221       !!x. x\<in>A ==> f x = g x |] ==> f = g"
```
```   222 by (force simp add: fun_eq_iff extensional_def)
```
```   223
```
```   224 lemma extensional_restrict:  "f \<in> extensional A \<Longrightarrow> restrict f A = f"
```
```   225 by(rule extensionalityI[OF restrict_extensional]) auto
```
```   226
```
```   227 lemma inv_into_funcset: "f ` A = B ==> (\<lambda>x\<in>B. inv_into A f x) : B -> A"
```
```   228 by (unfold inv_into_def) (fast intro: someI2)
```
```   229
```
```   230 lemma compose_inv_into_id:
```
```   231   "bij_betw f A B ==> compose A (\<lambda>y\<in>B. inv_into A f y) f = (\<lambda>x\<in>A. x)"
```
```   232 apply (simp add: bij_betw_def compose_def)
```
```   233 apply (rule restrict_ext, auto)
```
```   234 done
```
```   235
```
```   236 lemma compose_id_inv_into:
```
```   237   "f ` A = B ==> compose B f (\<lambda>y\<in>B. inv_into A f y) = (\<lambda>x\<in>B. x)"
```
```   238 apply (simp add: compose_def)
```
```   239 apply (rule restrict_ext)
```
```   240 apply (simp add: f_inv_into_f)
```
```   241 done
```
```   242
```
```   243
```
```   244 subsection{*Cardinality*}
```
```   245
```
```   246 lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
```
```   247 by (rule card_inj_on_le) auto
```
```   248
```
```   249 lemma card_bij:
```
```   250   "[|f \<in> A\<rightarrow>B; inj_on f A;
```
```   251      g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
```
```   252 by (blast intro: card_inj order_antisym)
```
```   253
```
```   254 end
```