1.1 --- a/src/HOL/Complete_Lattice.thy Wed Dec 08 14:52:23 2010 +0100
1.2 +++ b/src/HOL/Complete_Lattice.thy Wed Dec 08 15:05:46 2010 +0100
1.3 @@ -82,11 +82,21 @@
1.4 "\<Squnion>{a, b} = a \<squnion> b"
1.5 by (simp add: Sup_empty Sup_insert)
1.6
1.7 +lemma le_Inf_iff: "b \<sqsubseteq> Inf A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
1.8 + by (auto intro: Inf_greatest dest: Inf_lower)
1.9 +
1.10 lemma Sup_le_iff: "Sup A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
1.11 by (auto intro: Sup_least dest: Sup_upper)
1.12
1.13 -lemma le_Inf_iff: "b \<sqsubseteq> Inf A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
1.14 - by (auto intro: Inf_greatest dest: Inf_lower)
1.15 +lemma Inf_mono:
1.16 + assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<le> b"
1.17 + shows "Inf A \<le> Inf B"
1.18 +proof (rule Inf_greatest)
1.19 + fix b assume "b \<in> B"
1.20 + with assms obtain a where "a \<in> A" and "a \<le> b" by blast
1.21 + from `a \<in> A` have "Inf A \<le> a" by (rule Inf_lower)
1.22 + with `a \<le> b` show "Inf A \<le> b" by auto
1.23 +qed
1.24
1.25 lemma Sup_mono:
1.26 assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<le> b"
1.27 @@ -98,49 +108,39 @@
1.28 with `a \<le> b` show "a \<le> Sup B" by auto
1.29 qed
1.30
1.31 -lemma Inf_mono:
1.32 - assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<le> b"
1.33 - shows "Inf A \<le> Inf B"
1.34 -proof (rule Inf_greatest)
1.35 - fix b assume "b \<in> B"
1.36 - with assms obtain a where "a \<in> A" and "a \<le> b" by blast
1.37 - from `a \<in> A` have "Inf A \<le> a" by (rule Inf_lower)
1.38 - with `a \<le> b` show "Inf A \<le> b" by auto
1.39 -qed
1.40 +definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
1.41 + "INFI A f = \<Sqinter> (f ` A)"
1.42
1.43 definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
1.44 "SUPR A f = \<Squnion> (f ` A)"
1.45
1.46 -definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
1.47 - "INFI A f = \<Sqinter> (f ` A)"
1.48 -
1.49 end
1.50
1.51 syntax
1.52 + "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _./ _)" [0, 10] 10)
1.53 + "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _:_./ _)" [0, 0, 10] 10)
1.54 "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _./ _)" [0, 10] 10)
1.55 "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _:_./ _)" [0, 0, 10] 10)
1.56 - "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _./ _)" [0, 10] 10)
1.57 - "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _:_./ _)" [0, 0, 10] 10)
1.58
1.59 syntax (xsymbols)
1.60 + "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
1.61 + "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
1.62 "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10)
1.63 "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
1.64 - "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
1.65 - "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
1.66
1.67 translations
1.68 + "INF x y. B" == "INF x. INF y. B"
1.69 + "INF x. B" == "CONST INFI CONST UNIV (%x. B)"
1.70 + "INF x. B" == "INF x:CONST UNIV. B"
1.71 + "INF x:A. B" == "CONST INFI A (%x. B)"
1.72 "SUP x y. B" == "SUP x. SUP y. B"
1.73 "SUP x. B" == "CONST SUPR CONST UNIV (%x. B)"
1.74 "SUP x. B" == "SUP x:CONST UNIV. B"
1.75 "SUP x:A. B" == "CONST SUPR A (%x. B)"
1.76 - "INF x y. B" == "INF x. INF y. B"
1.77 - "INF x. B" == "CONST INFI CONST UNIV (%x. B)"
1.78 - "INF x. B" == "INF x:CONST UNIV. B"
1.79 - "INF x:A. B" == "CONST INFI A (%x. B)"
1.80
1.81 print_translation {*
1.82 - [Syntax.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"},
1.83 - Syntax.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"}]
1.84 + [Syntax.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
1.85 + Syntax.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
1.86 *} -- {* to avoid eta-contraction of body *}
1.87
1.88 context complete_lattice
1.89 @@ -164,54 +164,54 @@
1.90 lemma le_INF_iff: "u \<sqsubseteq> (INF i:A. M i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> M i)"
1.91 unfolding INFI_def by (auto simp add: le_Inf_iff)
1.92
1.93 +lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
1.94 + by (auto intro: antisym INF_leI le_INFI)
1.95 +
1.96 lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
1.97 by (auto intro: antisym SUP_leI le_SUPI)
1.98
1.99 -lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
1.100 - by (auto intro: antisym INF_leI le_INFI)
1.101 +lemma INF_mono:
1.102 + "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<le> g m) \<Longrightarrow> (INF n:A. f n) \<le> (INF n:B. g n)"
1.103 + by (force intro!: Inf_mono simp: INFI_def)
1.104
1.105 lemma SUP_mono:
1.106 "(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<le> g m) \<Longrightarrow> (SUP n:A. f n) \<le> (SUP n:B. g n)"
1.107 by (force intro!: Sup_mono simp: SUPR_def)
1.108
1.109 -lemma INF_mono:
1.110 - "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<le> g m) \<Longrightarrow> (INF n:A. f n) \<le> (INF n:B. g n)"
1.111 - by (force intro!: Inf_mono simp: INFI_def)
1.112 +lemma INF_subset: "A \<subseteq> B \<Longrightarrow> INFI B f \<le> INFI A f"
1.113 + by (intro INF_mono) auto
1.114
1.115 lemma SUP_subset: "A \<subseteq> B \<Longrightarrow> SUPR A f \<le> SUPR B f"
1.116 by (intro SUP_mono) auto
1.117
1.118 -lemma INF_subset: "A \<subseteq> B \<Longrightarrow> INFI B f \<le> INFI A f"
1.119 - by (intro INF_mono) auto
1.120 +lemma INF_commute: "(INF i:A. INF j:B. f i j) = (INF j:B. INF i:A. f i j)"
1.121 + by (iprover intro: INF_leI le_INFI order_trans antisym)
1.122
1.123 lemma SUP_commute: "(SUP i:A. SUP j:B. f i j) = (SUP j:B. SUP i:A. f i j)"
1.124 by (iprover intro: SUP_leI le_SUPI order_trans antisym)
1.125
1.126 -lemma INF_commute: "(INF i:A. INF j:B. f i j) = (INF j:B. INF i:A. f i j)"
1.127 - by (iprover intro: INF_leI le_INFI order_trans antisym)
1.128 +end
1.129
1.130 -end
1.131 +lemma Inf_less_iff:
1.132 + fixes a :: "'a\<Colon>{complete_lattice,linorder}"
1.133 + shows "Inf S < a \<longleftrightarrow> (\<exists>x\<in>S. x < a)"
1.134 + unfolding not_le[symmetric] le_Inf_iff by auto
1.135
1.136 lemma less_Sup_iff:
1.137 fixes a :: "'a\<Colon>{complete_lattice,linorder}"
1.138 shows "a < Sup S \<longleftrightarrow> (\<exists>x\<in>S. a < x)"
1.139 unfolding not_le[symmetric] Sup_le_iff by auto
1.140
1.141 -lemma Inf_less_iff:
1.142 - fixes a :: "'a\<Colon>{complete_lattice,linorder}"
1.143 - shows "Inf S < a \<longleftrightarrow> (\<exists>x\<in>S. x < a)"
1.144 - unfolding not_le[symmetric] le_Inf_iff by auto
1.145 +lemma INF_less_iff:
1.146 + fixes a :: "'a::{complete_lattice,linorder}"
1.147 + shows "(INF i:A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)"
1.148 + unfolding INFI_def Inf_less_iff by auto
1.149
1.150 lemma less_SUP_iff:
1.151 fixes a :: "'a::{complete_lattice,linorder}"
1.152 shows "a < (SUP i:A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < f x)"
1.153 unfolding SUPR_def less_Sup_iff by auto
1.154
1.155 -lemma INF_less_iff:
1.156 - fixes a :: "'a::{complete_lattice,linorder}"
1.157 - shows "(INF i:A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)"
1.158 - unfolding INFI_def Inf_less_iff by auto
1.159 -
1.160 subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
1.161
1.162 instantiation bool :: complete_lattice
1.163 @@ -278,6 +278,200 @@
1.164 by (auto intro: arg_cong [of _ _ Sup] simp add: SUPR_def Sup_apply)
1.165
1.166
1.167 +subsection {* Inter *}
1.168 +
1.169 +abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
1.170 + "Inter S \<equiv> \<Sqinter>S"
1.171 +
1.172 +notation (xsymbols)
1.173 + Inter ("\<Inter>_" [90] 90)
1.174 +
1.175 +lemma Inter_eq:
1.176 + "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
1.177 +proof (rule set_eqI)
1.178 + fix x
1.179 + have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
1.180 + by auto
1.181 + then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
1.182 + by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
1.183 +qed
1.184 +
1.185 +lemma Inter_iff [simp,no_atp]: "(A : Inter C) = (ALL X:C. A:X)"
1.186 + by (unfold Inter_eq) blast
1.187 +
1.188 +lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
1.189 + by (simp add: Inter_eq)
1.190 +
1.191 +text {*
1.192 + \medskip A ``destruct'' rule -- every @{term X} in @{term C}
1.193 + contains @{term A} as an element, but @{prop "A:X"} can hold when
1.194 + @{prop "X:C"} does not! This rule is analogous to @{text spec}.
1.195 +*}
1.196 +
1.197 +lemma InterD [elim, Pure.elim]: "A : Inter C ==> X:C ==> A:X"
1.198 + by auto
1.199 +
1.200 +lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
1.201 + -- {* ``Classical'' elimination rule -- does not require proving
1.202 + @{prop "X:C"}. *}
1.203 + by (unfold Inter_eq) blast
1.204 +
1.205 +lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
1.206 + by blast
1.207 +
1.208 +lemma Inter_subset:
1.209 + "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
1.210 + by blast
1.211 +
1.212 +lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
1.213 + by (iprover intro: InterI subsetI dest: subsetD)
1.214 +
1.215 +lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
1.216 + by blast
1.217 +
1.218 +lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
1.219 + by (fact Inf_empty)
1.220 +
1.221 +lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
1.222 + by blast
1.223 +
1.224 +lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
1.225 + by blast
1.226 +
1.227 +lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
1.228 + by blast
1.229 +
1.230 +lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
1.231 + by blast
1.232 +
1.233 +lemma Inter_UNIV_conv [simp,no_atp]:
1.234 + "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
1.235 + "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
1.236 + by blast+
1.237 +
1.238 +lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"
1.239 + by blast
1.240 +
1.241 +
1.242 +subsection {* Intersections of families *}
1.243 +
1.244 +abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
1.245 + "INTER \<equiv> INFI"
1.246 +
1.247 +syntax
1.248 + "_INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10)
1.249 + "_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 0, 10] 10)
1.250 +
1.251 +syntax (xsymbols)
1.252 + "_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10)
1.253 + "_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
1.254 +
1.255 +syntax (latex output)
1.256 + "_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
1.257 + "_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
1.258 +
1.259 +translations
1.260 + "INT x y. B" == "INT x. INT y. B"
1.261 + "INT x. B" == "CONST INTER CONST UNIV (%x. B)"
1.262 + "INT x. B" == "INT x:CONST UNIV. B"
1.263 + "INT x:A. B" == "CONST INTER A (%x. B)"
1.264 +
1.265 +print_translation {*
1.266 + [Syntax.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
1.267 +*} -- {* to avoid eta-contraction of body *}
1.268 +
1.269 +lemma INTER_eq_Inter_image:
1.270 + "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
1.271 + by (fact INFI_def)
1.272 +
1.273 +lemma Inter_def:
1.274 + "\<Inter>S = (\<Inter>x\<in>S. x)"
1.275 + by (simp add: INTER_eq_Inter_image image_def)
1.276 +
1.277 +lemma INTER_def:
1.278 + "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
1.279 + by (auto simp add: INTER_eq_Inter_image Inter_eq)
1.280 +
1.281 +lemma Inter_image_eq [simp]:
1.282 + "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
1.283 + by (rule sym) (fact INTER_eq_Inter_image)
1.284 +
1.285 +lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
1.286 + by (unfold INTER_def) blast
1.287 +
1.288 +lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
1.289 + by (unfold INTER_def) blast
1.290 +
1.291 +lemma INT_D [elim, Pure.elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
1.292 + by auto
1.293 +
1.294 +lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
1.295 + -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
1.296 + by (unfold INTER_def) blast
1.297 +
1.298 +lemma INT_cong [cong]:
1.299 + "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
1.300 + by (simp add: INTER_def)
1.301 +
1.302 +lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
1.303 + by blast
1.304 +
1.305 +lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
1.306 + by blast
1.307 +
1.308 +lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
1.309 + by (fact INF_leI)
1.310 +
1.311 +lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
1.312 + by (fact le_INFI)
1.313 +
1.314 +lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
1.315 + by blast
1.316 +
1.317 +lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
1.318 + by blast
1.319 +
1.320 +lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
1.321 + by (fact le_INF_iff)
1.322 +
1.323 +lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
1.324 + by blast
1.325 +
1.326 +lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
1.327 + by blast
1.328 +
1.329 +lemma INT_insert_distrib:
1.330 + "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
1.331 + by blast
1.332 +
1.333 +lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
1.334 + by auto
1.335 +
1.336 +lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
1.337 + -- {* Look: it has an \emph{existential} quantifier *}
1.338 + by blast
1.339 +
1.340 +lemma INTER_UNIV_conv[simp]:
1.341 + "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
1.342 + "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
1.343 +by blast+
1.344 +
1.345 +lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
1.346 + by (auto intro: bool_induct)
1.347 +
1.348 +lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
1.349 + by blast
1.350 +
1.351 +lemma INT_anti_mono:
1.352 + "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
1.353 + (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
1.354 + -- {* The last inclusion is POSITIVE! *}
1.355 + by (blast dest: subsetD)
1.356 +
1.357 +lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
1.358 + by blast
1.359 +
1.360 +
1.361 subsection {* Union *}
1.362
1.363 abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
1.364 @@ -514,200 +708,6 @@
1.365 by blast
1.366
1.367
1.368 -subsection {* Inter *}
1.369 -
1.370 -abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
1.371 - "Inter S \<equiv> \<Sqinter>S"
1.372 -
1.373 -notation (xsymbols)
1.374 - Inter ("\<Inter>_" [90] 90)
1.375 -
1.376 -lemma Inter_eq:
1.377 - "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
1.378 -proof (rule set_eqI)
1.379 - fix x
1.380 - have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
1.381 - by auto
1.382 - then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
1.383 - by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
1.384 -qed
1.385 -
1.386 -lemma Inter_iff [simp,no_atp]: "(A : Inter C) = (ALL X:C. A:X)"
1.387 - by (unfold Inter_eq) blast
1.388 -
1.389 -lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
1.390 - by (simp add: Inter_eq)
1.391 -
1.392 -text {*
1.393 - \medskip A ``destruct'' rule -- every @{term X} in @{term C}
1.394 - contains @{term A} as an element, but @{prop "A:X"} can hold when
1.395 - @{prop "X:C"} does not! This rule is analogous to @{text spec}.
1.396 -*}
1.397 -
1.398 -lemma InterD [elim, Pure.elim]: "A : Inter C ==> X:C ==> A:X"
1.399 - by auto
1.400 -
1.401 -lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
1.402 - -- {* ``Classical'' elimination rule -- does not require proving
1.403 - @{prop "X:C"}. *}
1.404 - by (unfold Inter_eq) blast
1.405 -
1.406 -lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
1.407 - by blast
1.408 -
1.409 -lemma Inter_subset:
1.410 - "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
1.411 - by blast
1.412 -
1.413 -lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
1.414 - by (iprover intro: InterI subsetI dest: subsetD)
1.415 -
1.416 -lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
1.417 - by blast
1.418 -
1.419 -lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
1.420 - by (fact Inf_empty)
1.421 -
1.422 -lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
1.423 - by blast
1.424 -
1.425 -lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
1.426 - by blast
1.427 -
1.428 -lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
1.429 - by blast
1.430 -
1.431 -lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
1.432 - by blast
1.433 -
1.434 -lemma Inter_UNIV_conv [simp,no_atp]:
1.435 - "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
1.436 - "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
1.437 - by blast+
1.438 -
1.439 -lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"
1.440 - by blast
1.441 -
1.442 -
1.443 -subsection {* Intersections of families *}
1.444 -
1.445 -abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
1.446 - "INTER \<equiv> INFI"
1.447 -
1.448 -syntax
1.449 - "_INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10)
1.450 - "_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 0, 10] 10)
1.451 -
1.452 -syntax (xsymbols)
1.453 - "_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10)
1.454 - "_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
1.455 -
1.456 -syntax (latex output)
1.457 - "_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
1.458 - "_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
1.459 -
1.460 -translations
1.461 - "INT x y. B" == "INT x. INT y. B"
1.462 - "INT x. B" == "CONST INTER CONST UNIV (%x. B)"
1.463 - "INT x. B" == "INT x:CONST UNIV. B"
1.464 - "INT x:A. B" == "CONST INTER A (%x. B)"
1.465 -
1.466 -print_translation {*
1.467 - [Syntax.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
1.468 -*} -- {* to avoid eta-contraction of body *}
1.469 -
1.470 -lemma INTER_eq_Inter_image:
1.471 - "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
1.472 - by (fact INFI_def)
1.473 -
1.474 -lemma Inter_def:
1.475 - "\<Inter>S = (\<Inter>x\<in>S. x)"
1.476 - by (simp add: INTER_eq_Inter_image image_def)
1.477 -
1.478 -lemma INTER_def:
1.479 - "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
1.480 - by (auto simp add: INTER_eq_Inter_image Inter_eq)
1.481 -
1.482 -lemma Inter_image_eq [simp]:
1.483 - "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
1.484 - by (rule sym) (fact INTER_eq_Inter_image)
1.485 -
1.486 -lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
1.487 - by (unfold INTER_def) blast
1.488 -
1.489 -lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
1.490 - by (unfold INTER_def) blast
1.491 -
1.492 -lemma INT_D [elim, Pure.elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
1.493 - by auto
1.494 -
1.495 -lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
1.496 - -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
1.497 - by (unfold INTER_def) blast
1.498 -
1.499 -lemma INT_cong [cong]:
1.500 - "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
1.501 - by (simp add: INTER_def)
1.502 -
1.503 -lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
1.504 - by blast
1.505 -
1.506 -lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
1.507 - by blast
1.508 -
1.509 -lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
1.510 - by (fact INF_leI)
1.511 -
1.512 -lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
1.513 - by (fact le_INFI)
1.514 -
1.515 -lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
1.516 - by blast
1.517 -
1.518 -lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
1.519 - by blast
1.520 -
1.521 -lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
1.522 - by (fact le_INF_iff)
1.523 -
1.524 -lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
1.525 - by blast
1.526 -
1.527 -lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
1.528 - by blast
1.529 -
1.530 -lemma INT_insert_distrib:
1.531 - "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
1.532 - by blast
1.533 -
1.534 -lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
1.535 - by auto
1.536 -
1.537 -lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
1.538 - -- {* Look: it has an \emph{existential} quantifier *}
1.539 - by blast
1.540 -
1.541 -lemma INTER_UNIV_conv[simp]:
1.542 - "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
1.543 - "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
1.544 -by blast+
1.545 -
1.546 -lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
1.547 - by (auto intro: bool_induct)
1.548 -
1.549 -lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
1.550 - by blast
1.551 -
1.552 -lemma INT_anti_mono:
1.553 - "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
1.554 - (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
1.555 - -- {* The last inclusion is POSITIVE! *}
1.556 - by (blast dest: subsetD)
1.557 -
1.558 -lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
1.559 - by blast
1.560 -
1.561 -
1.562 subsection {* Distributive laws *}
1.563
1.564 lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
1.565 @@ -858,18 +858,18 @@
1.566 no_notation
1.567 less_eq (infix "\<sqsubseteq>" 50) and
1.568 less (infix "\<sqsubset>" 50) and
1.569 + bot ("\<bottom>") and
1.570 + top ("\<top>") and
1.571 inf (infixl "\<sqinter>" 70) and
1.572 sup (infixl "\<squnion>" 65) and
1.573 Inf ("\<Sqinter>_" [900] 900) and
1.574 - Sup ("\<Squnion>_" [900] 900) and
1.575 - top ("\<top>") and
1.576 - bot ("\<bottom>")
1.577 + Sup ("\<Squnion>_" [900] 900)
1.578
1.579 no_syntax (xsymbols)
1.580 + "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
1.581 + "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
1.582 "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10)
1.583 "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
1.584 - "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
1.585 - "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
1.586
1.587 lemmas mem_simps =
1.588 insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
2.1 --- a/src/HOL/Lattices.thy Wed Dec 08 14:52:23 2010 +0100
2.2 +++ b/src/HOL/Lattices.thy Wed Dec 08 15:05:46 2010 +0100
2.3 @@ -48,8 +48,9 @@
2.4 notation
2.5 less_eq (infix "\<sqsubseteq>" 50) and
2.6 less (infix "\<sqsubset>" 50) and
2.7 - top ("\<top>") and
2.8 - bot ("\<bottom>")
2.9 + bot ("\<bottom>") and
2.10 + top ("\<top>")
2.11 +
2.12
2.13 class semilattice_inf = order +
2.14 fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
3.1 --- a/src/HOL/Library/Lattice_Syntax.thy Wed Dec 08 14:52:23 2010 +0100
3.2 +++ b/src/HOL/Library/Lattice_Syntax.thy Wed Dec 08 15:05:46 2010 +0100
3.3 @@ -7,17 +7,17 @@
3.4 begin
3.5
3.6 notation
3.7 + bot ("\<bottom>") and
3.8 top ("\<top>") and
3.9 - bot ("\<bottom>") and
3.10 inf (infixl "\<sqinter>" 70) and
3.11 sup (infixl "\<squnion>" 65) and
3.12 Inf ("\<Sqinter>_" [900] 900) and
3.13 Sup ("\<Squnion>_" [900] 900)
3.14
3.15 syntax (xsymbols)
3.16 + "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
3.17 + "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
3.18 "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10)
3.19 "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
3.20 - "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
3.21 - "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
3.22
3.23 end
4.1 --- a/src/HOL/Orderings.thy Wed Dec 08 14:52:23 2010 +0100
4.2 +++ b/src/HOL/Orderings.thy Wed Dec 08 15:05:46 2010 +0100
4.3 @@ -1082,14 +1082,14 @@
4.4
4.5 subsection {* Top and bottom elements *}
4.6
4.7 +class bot = preorder +
4.8 + fixes bot :: 'a
4.9 + assumes bot_least [simp]: "bot \<le> x"
4.10 +
4.11 class top = preorder +
4.12 fixes top :: 'a
4.13 assumes top_greatest [simp]: "x \<le> top"
4.14
4.15 -class bot = preorder +
4.16 - fixes bot :: 'a
4.17 - assumes bot_least [simp]: "bot \<le> x"
4.18 -
4.19
4.20 subsection {* Dense orders *}
4.21
4.22 @@ -1204,7 +1204,7 @@
4.23
4.24 subsection {* Order on bool *}
4.25
4.26 -instantiation bool :: "{order, top, bot}"
4.27 +instantiation bool :: "{order, bot, top}"
4.28 begin
4.29
4.30 definition
4.31 @@ -1214,10 +1214,10 @@
4.32 [simp]: "(P\<Colon>bool) < Q \<longleftrightarrow> \<not> P \<and> Q"
4.33
4.34 definition
4.35 - [simp]: "top \<longleftrightarrow> True"
4.36 + [simp]: "bot \<longleftrightarrow> False"
4.37
4.38 definition
4.39 - [simp]: "bot \<longleftrightarrow> False"
4.40 + [simp]: "top \<longleftrightarrow> True"
4.41
4.42 instance proof
4.43 qed auto
4.44 @@ -1272,6 +1272,21 @@
4.45 instance "fun" :: (type, order) order proof
4.46 qed (auto simp add: le_fun_def intro: antisym ext)
4.47
4.48 +instantiation "fun" :: (type, bot) bot
4.49 +begin
4.50 +
4.51 +definition
4.52 + "bot = (\<lambda>x. bot)"
4.53 +
4.54 +lemma bot_apply:
4.55 + "bot x = bot"
4.56 + by (simp add: bot_fun_def)
4.57 +
4.58 +instance proof
4.59 +qed (simp add: le_fun_def bot_apply)
4.60 +
4.61 +end
4.62 +
4.63 instantiation "fun" :: (type, top) top
4.64 begin
4.65
4.66 @@ -1288,21 +1303,6 @@
4.67
4.68 end
4.69
4.70 -instantiation "fun" :: (type, bot) bot
4.71 -begin
4.72 -
4.73 -definition
4.74 - "bot = (\<lambda>x. bot)"
4.75 -
4.76 -lemma bot_apply:
4.77 - "bot x = bot"
4.78 - by (simp add: bot_fun_def)
4.79 -
4.80 -instance proof
4.81 -qed (simp add: le_fun_def bot_apply)
4.82 -
4.83 -end
4.84 -
4.85 lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g"
4.86 unfolding le_fun_def by simp
4.87
5.1 --- a/src/HOL/Predicate.thy Wed Dec 08 14:52:23 2010 +0100
5.2 +++ b/src/HOL/Predicate.thy Wed Dec 08 15:05:46 2010 +0100
5.3 @@ -9,18 +9,18 @@
5.4 begin
5.5
5.6 notation
5.7 + bot ("\<bottom>") and
5.8 + top ("\<top>") and
5.9 inf (infixl "\<sqinter>" 70) and
5.10 sup (infixl "\<squnion>" 65) and
5.11 Inf ("\<Sqinter>_" [900] 900) and
5.12 - Sup ("\<Squnion>_" [900] 900) and
5.13 - top ("\<top>") and
5.14 - bot ("\<bottom>")
5.15 + Sup ("\<Squnion>_" [900] 900)
5.16
5.17 syntax (xsymbols)
5.18 + "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
5.19 + "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
5.20 "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10)
5.21 "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
5.22 - "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
5.23 - "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
5.24
5.25
5.26 subsection {* Predicates as (complete) lattices *}
5.27 @@ -92,12 +92,6 @@
5.28
5.29 subsubsection {* Top and bottom elements *}
5.30
5.31 -lemma top1I [intro!]: "top x"
5.32 - by (simp add: top_fun_def top_bool_def)
5.33 -
5.34 -lemma top2I [intro!]: "top x y"
5.35 - by (simp add: top_fun_def top_bool_def)
5.36 -
5.37 lemma bot1E [no_atp, elim!]: "bot x \<Longrightarrow> P"
5.38 by (simp add: bot_fun_def bot_bool_def)
5.39
5.40 @@ -110,6 +104,45 @@
5.41 lemma bot_empty_eq2: "bot = (\<lambda>x y. (x, y) \<in> {})"
5.42 by (auto simp add: fun_eq_iff)
5.43
5.44 +lemma top1I [intro!]: "top x"
5.45 + by (simp add: top_fun_def top_bool_def)
5.46 +
5.47 +lemma top2I [intro!]: "top x y"
5.48 + by (simp add: top_fun_def top_bool_def)
5.49 +
5.50 +
5.51 +subsubsection {* Binary intersection *}
5.52 +
5.53 +lemma inf1I [intro!]: "A x ==> B x ==> inf A B x"
5.54 + by (simp add: inf_fun_def inf_bool_def)
5.55 +
5.56 +lemma inf2I [intro!]: "A x y ==> B x y ==> inf A B x y"
5.57 + by (simp add: inf_fun_def inf_bool_def)
5.58 +
5.59 +lemma inf1E [elim!]: "inf A B x ==> (A x ==> B x ==> P) ==> P"
5.60 + by (simp add: inf_fun_def inf_bool_def)
5.61 +
5.62 +lemma inf2E [elim!]: "inf A B x y ==> (A x y ==> B x y ==> P) ==> P"
5.63 + by (simp add: inf_fun_def inf_bool_def)
5.64 +
5.65 +lemma inf1D1: "inf A B x ==> A x"
5.66 + by (simp add: inf_fun_def inf_bool_def)
5.67 +
5.68 +lemma inf2D1: "inf A B x y ==> A x y"
5.69 + by (simp add: inf_fun_def inf_bool_def)
5.70 +
5.71 +lemma inf1D2: "inf A B x ==> B x"
5.72 + by (simp add: inf_fun_def inf_bool_def)
5.73 +
5.74 +lemma inf2D2: "inf A B x y ==> B x y"
5.75 + by (simp add: inf_fun_def inf_bool_def)
5.76 +
5.77 +lemma inf_Int_eq: "inf (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
5.78 + by (simp add: inf_fun_def inf_bool_def mem_def)
5.79 +
5.80 +lemma inf_Int_eq2 [pred_set_conv]: "inf (\<lambda>x y. (x, y) \<in> R) (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)"
5.81 + by (simp add: inf_fun_def inf_bool_def mem_def)
5.82 +
5.83
5.84 subsubsection {* Binary union *}
5.85
5.86 @@ -149,66 +182,6 @@
5.87 by (simp add: sup_fun_def sup_bool_def mem_def)
5.88
5.89
5.90 -subsubsection {* Binary intersection *}
5.91 -
5.92 -lemma inf1I [intro!]: "A x ==> B x ==> inf A B x"
5.93 - by (simp add: inf_fun_def inf_bool_def)
5.94 -
5.95 -lemma inf2I [intro!]: "A x y ==> B x y ==> inf A B x y"
5.96 - by (simp add: inf_fun_def inf_bool_def)
5.97 -
5.98 -lemma inf1E [elim!]: "inf A B x ==> (A x ==> B x ==> P) ==> P"
5.99 - by (simp add: inf_fun_def inf_bool_def)
5.100 -
5.101 -lemma inf2E [elim!]: "inf A B x y ==> (A x y ==> B x y ==> P) ==> P"
5.102 - by (simp add: inf_fun_def inf_bool_def)
5.103 -
5.104 -lemma inf1D1: "inf A B x ==> A x"
5.105 - by (simp add: inf_fun_def inf_bool_def)
5.106 -
5.107 -lemma inf2D1: "inf A B x y ==> A x y"
5.108 - by (simp add: inf_fun_def inf_bool_def)
5.109 -
5.110 -lemma inf1D2: "inf A B x ==> B x"
5.111 - by (simp add: inf_fun_def inf_bool_def)
5.112 -
5.113 -lemma inf2D2: "inf A B x y ==> B x y"
5.114 - by (simp add: inf_fun_def inf_bool_def)
5.115 -
5.116 -lemma inf_Int_eq: "inf (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
5.117 - by (simp add: inf_fun_def inf_bool_def mem_def)
5.118 -
5.119 -lemma inf_Int_eq2 [pred_set_conv]: "inf (\<lambda>x y. (x, y) \<in> R) (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)"
5.120 - by (simp add: inf_fun_def inf_bool_def mem_def)
5.121 -
5.122 -
5.123 -subsubsection {* Unions of families *}
5.124 -
5.125 -lemma SUP1_iff: "(SUP x:A. B x) b = (EX x:A. B x b)"
5.126 - by (simp add: SUPR_apply)
5.127 -
5.128 -lemma SUP2_iff: "(SUP x:A. B x) b c = (EX x:A. B x b c)"
5.129 - by (simp add: SUPR_apply)
5.130 -
5.131 -lemma SUP1_I [intro]: "a : A ==> B a b ==> (SUP x:A. B x) b"
5.132 - by (auto simp add: SUPR_apply)
5.133 -
5.134 -lemma SUP2_I [intro]: "a : A ==> B a b c ==> (SUP x:A. B x) b c"
5.135 - by (auto simp add: SUPR_apply)
5.136 -
5.137 -lemma SUP1_E [elim!]: "(SUP x:A. B x) b ==> (!!x. x : A ==> B x b ==> R) ==> R"
5.138 - by (auto simp add: SUPR_apply)
5.139 -
5.140 -lemma SUP2_E [elim!]: "(SUP x:A. B x) b c ==> (!!x. x : A ==> B x b c ==> R) ==> R"
5.141 - by (auto simp add: SUPR_apply)
5.142 -
5.143 -lemma SUP_UN_eq: "(SUP i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (UN i. r i))"
5.144 - by (simp add: SUPR_apply fun_eq_iff)
5.145 -
5.146 -lemma SUP_UN_eq2: "(SUP i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (UN i. r i))"
5.147 - by (simp add: SUPR_apply fun_eq_iff)
5.148 -
5.149 -
5.150 subsubsection {* Intersections of families *}
5.151
5.152 lemma INF1_iff: "(INF x:A. B x) b = (ALL x:A. B x b)"
5.153 @@ -242,6 +215,33 @@
5.154 by (simp add: INFI_apply fun_eq_iff)
5.155
5.156
5.157 +subsubsection {* Unions of families *}
5.158 +
5.159 +lemma SUP1_iff: "(SUP x:A. B x) b = (EX x:A. B x b)"
5.160 + by (simp add: SUPR_apply)
5.161 +
5.162 +lemma SUP2_iff: "(SUP x:A. B x) b c = (EX x:A. B x b c)"
5.163 + by (simp add: SUPR_apply)
5.164 +
5.165 +lemma SUP1_I [intro]: "a : A ==> B a b ==> (SUP x:A. B x) b"
5.166 + by (auto simp add: SUPR_apply)
5.167 +
5.168 +lemma SUP2_I [intro]: "a : A ==> B a b c ==> (SUP x:A. B x) b c"
5.169 + by (auto simp add: SUPR_apply)
5.170 +
5.171 +lemma SUP1_E [elim!]: "(SUP x:A. B x) b ==> (!!x. x : A ==> B x b ==> R) ==> R"
5.172 + by (auto simp add: SUPR_apply)
5.173 +
5.174 +lemma SUP2_E [elim!]: "(SUP x:A. B x) b c ==> (!!x. x : A ==> B x b c ==> R) ==> R"
5.175 + by (auto simp add: SUPR_apply)
5.176 +
5.177 +lemma SUP_UN_eq: "(SUP i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (UN i. r i))"
5.178 + by (simp add: SUPR_apply fun_eq_iff)
5.179 +
5.180 +lemma SUP_UN_eq2: "(SUP i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (UN i. r i))"
5.181 + by (simp add: SUPR_apply fun_eq_iff)
5.182 +
5.183 +
5.184 subsection {* Predicates as relations *}
5.185
5.186 subsubsection {* Composition *}
5.187 @@ -1027,19 +1027,19 @@
5.188 *}
5.189
5.190 no_notation
5.191 + bot ("\<bottom>") and
5.192 + top ("\<top>") and
5.193 inf (infixl "\<sqinter>" 70) and
5.194 sup (infixl "\<squnion>" 65) and
5.195 Inf ("\<Sqinter>_" [900] 900) and
5.196 Sup ("\<Squnion>_" [900] 900) and
5.197 - top ("\<top>") and
5.198 - bot ("\<bottom>") and
5.199 bind (infixl "\<guillemotright>=" 70)
5.200
5.201 no_syntax (xsymbols)
5.202 + "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
5.203 + "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
5.204 "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10)
5.205 "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
5.206 - "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
5.207 - "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
5.208
5.209 hide_type (open) pred seq
5.210 hide_const (open) Pred eval single bind is_empty singleton if_pred not_pred holds
6.1 --- a/src/HOL/Set.thy Wed Dec 08 14:52:23 2010 +0100
6.2 +++ b/src/HOL/Set.thy Wed Dec 08 15:05:46 2010 +0100
6.3 @@ -533,6 +533,36 @@
6.4 by simp
6.5
6.6
6.7 +subsubsection {* The empty set *}
6.8 +
6.9 +lemma empty_def:
6.10 + "{} = {x. False}"
6.11 + by (simp add: bot_fun_def bot_bool_def Collect_def)
6.12 +
6.13 +lemma empty_iff [simp]: "(c : {}) = False"
6.14 + by (simp add: empty_def)
6.15 +
6.16 +lemma emptyE [elim!]: "a : {} ==> P"
6.17 + by simp
6.18 +
6.19 +lemma empty_subsetI [iff]: "{} \<subseteq> A"
6.20 + -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
6.21 + by blast
6.22 +
6.23 +lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
6.24 + by blast
6.25 +
6.26 +lemma equals0D: "A = {} ==> a \<notin> A"
6.27 + -- {* Use for reasoning about disjointness: @{text "A Int B = {}"} *}
6.28 + by blast
6.29 +
6.30 +lemma ball_empty [simp]: "Ball {} P = True"
6.31 + by (simp add: Ball_def)
6.32 +
6.33 +lemma bex_empty [simp]: "Bex {} P = False"
6.34 + by (simp add: Bex_def)
6.35 +
6.36 +
6.37 subsubsection {* The universal set -- UNIV *}
6.38
6.39 abbreviation UNIV :: "'a set" where
6.40 @@ -568,36 +598,6 @@
6.41 lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A"
6.42 by auto
6.43
6.44 -
6.45 -subsubsection {* The empty set *}
6.46 -
6.47 -lemma empty_def:
6.48 - "{} = {x. False}"
6.49 - by (simp add: bot_fun_def bot_bool_def Collect_def)
6.50 -
6.51 -lemma empty_iff [simp]: "(c : {}) = False"
6.52 - by (simp add: empty_def)
6.53 -
6.54 -lemma emptyE [elim!]: "a : {} ==> P"
6.55 - by simp
6.56 -
6.57 -lemma empty_subsetI [iff]: "{} \<subseteq> A"
6.58 - -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
6.59 - by blast
6.60 -
6.61 -lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}"
6.62 - by blast
6.63 -
6.64 -lemma equals0D: "A = {} ==> a \<notin> A"
6.65 - -- {* Use for reasoning about disjointness: @{text "A Int B = {}"} *}
6.66 - by blast
6.67 -
6.68 -lemma ball_empty [simp]: "Ball {} P = True"
6.69 - by (simp add: Ball_def)
6.70 -
6.71 -lemma bex_empty [simp]: "Bex {} P = False"
6.72 - by (simp add: Bex_def)
6.73 -
6.74 lemma UNIV_not_empty [iff]: "UNIV ~= {}"
6.75 by (blast elim: equalityE)
6.76
6.77 @@ -647,7 +647,41 @@
6.78 lemma Compl_eq: "- A = {x. ~ x : A}" by blast
6.79
6.80
6.81 -subsubsection {* Binary union -- Un *}
6.82 +subsubsection {* Binary intersection *}
6.83 +
6.84 +abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where
6.85 + "op Int \<equiv> inf"
6.86 +
6.87 +notation (xsymbols)
6.88 + inter (infixl "\<inter>" 70)
6.89 +
6.90 +notation (HTML output)
6.91 + inter (infixl "\<inter>" 70)
6.92 +
6.93 +lemma Int_def:
6.94 + "A \<inter> B = {x. x \<in> A \<and> x \<in> B}"
6.95 + by (simp add: inf_fun_def inf_bool_def Collect_def mem_def)
6.96 +
6.97 +lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
6.98 + by (unfold Int_def) blast
6.99 +
6.100 +lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
6.101 + by simp
6.102 +
6.103 +lemma IntD1: "c : A Int B ==> c:A"
6.104 + by simp
6.105 +
6.106 +lemma IntD2: "c : A Int B ==> c:B"
6.107 + by simp
6.108 +
6.109 +lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
6.110 + by simp
6.111 +
6.112 +lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
6.113 + by (fact mono_inf)
6.114 +
6.115 +
6.116 +subsubsection {* Binary union *}
6.117
6.118 abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where
6.119 "union \<equiv> sup"
6.120 @@ -689,40 +723,6 @@
6.121 by (fact mono_sup)
6.122
6.123
6.124 -subsubsection {* Binary intersection -- Int *}
6.125 -
6.126 -abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where
6.127 - "op Int \<equiv> inf"
6.128 -
6.129 -notation (xsymbols)
6.130 - inter (infixl "\<inter>" 70)
6.131 -
6.132 -notation (HTML output)
6.133 - inter (infixl "\<inter>" 70)
6.134 -
6.135 -lemma Int_def:
6.136 - "A \<inter> B = {x. x \<in> A \<and> x \<in> B}"
6.137 - by (simp add: inf_fun_def inf_bool_def Collect_def mem_def)
6.138 -
6.139 -lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
6.140 - by (unfold Int_def) blast
6.141 -
6.142 -lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
6.143 - by simp
6.144 -
6.145 -lemma IntD1: "c : A Int B ==> c:A"
6.146 - by simp
6.147 -
6.148 -lemma IntD2: "c : A Int B ==> c:B"
6.149 - by simp
6.150 -
6.151 -lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
6.152 - by simp
6.153 -
6.154 -lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
6.155 - by (fact mono_inf)
6.156 -
6.157 -
6.158 subsubsection {* Set difference *}
6.159
6.160 lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"