wneuper/isa: src/HOL/Complete_Lattice.thy@9ff94e7cc3b3

     1 (*  Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)

     3 header {* Complete lattices, with special focus on sets *}

     5 theory Complete_Lattice

     6 imports Set

     7 begin

     9 notation

    10   less_eq (infix "\<sqsubseteq>" 50) and

    11   less (infix "\<sqsubset>" 50) and

    12   inf (infixl "\<sqinter>" 70) and

    13   sup (infixl "\<squnion>" 65) and

    14   top ("\<top>") and

    15   bot ("\<bottom>")

    18 subsection {* Syntactic infimum and supremum operations *}

    20 class Inf =

    21   fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)

    23 class Sup =

    24   fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)

    26 subsection {* Abstract complete lattices *}

    28 class complete_lattice = bounded_lattice + Inf + Sup +

    29   assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"

    30      and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"

    31   assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"

    32      and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"

    33 begin

    35 lemma dual_complete_lattice:

    36   "class.complete_lattice Sup Inf (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom>"

    37   by (auto intro!: class.complete_lattice.intro dual_bounded_lattice)

    38     (unfold_locales, (fact bot_least top_greatest

    39         Sup_upper Sup_least Inf_lower Inf_greatest)+)

    41 lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"

    42   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)

    44 lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"

    45   by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)

    47 lemma Inf_empty [simp]:

    48   "\<Sqinter>{} = \<top>"

    49   by (auto intro: antisym Inf_greatest)

    51 lemma Sup_empty [simp]:

    52   "\<Squnion>{} = \<bottom>"

    53   by (auto intro: antisym Sup_least)

    55 lemma Inf_UNIV [simp]:

    56   "\<Sqinter>UNIV = \<bottom>"

    57   by (simp add: Sup_Inf Sup_empty [symmetric])

    59 lemma Sup_UNIV [simp]:

    60   "\<Squnion>UNIV = \<top>"

    61   by (simp add: Inf_Sup Inf_empty [symmetric])

    63 lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"

    64   by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)

    66 lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"

    67   by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)

    69 lemma Inf_singleton [simp]:

    70   "\<Sqinter>{a} = a"

    71   by (auto intro: antisym Inf_lower Inf_greatest)

    73 lemma Sup_singleton [simp]:

    74   "\<Squnion>{a} = a"

    75   by (auto intro: antisym Sup_upper Sup_least)

    77 lemma Inf_binary:

    78   "\<Sqinter>{a, b} = a \<sqinter> b"

    79   by (simp add: Inf_empty Inf_insert)

    81 lemma Sup_binary:

    82   "\<Squnion>{a, b} = a \<squnion> b"

    83   by (simp add: Sup_empty Sup_insert)

    85 lemma le_Inf_iff: "b \<sqsubseteq> Inf A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"

    86   by (auto intro: Inf_greatest dest: Inf_lower)

    88 lemma Sup_le_iff: "Sup A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"

    89   by (auto intro: Sup_least dest: Sup_upper)

    91 lemma Inf_mono:

    92   assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<le> b"

    93   shows "Inf A \<le> Inf B"

    94 proof (rule Inf_greatest)

    95   fix b assume "b \<in> B"

    96   with assms obtain a where "a \<in> A" and "a \<le> b" by blast

    97   from `a \<in> A` have "Inf A \<le> a" by (rule Inf_lower)

    98   with `a \<le> b` show "Inf A \<le> b" by auto

    99 qed

   101 lemma Sup_mono:

   102   assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<le> b"

   103   shows "Sup A \<le> Sup B"

   104 proof (rule Sup_least)

   105   fix a assume "a \<in> A"

   106   with assms obtain b where "b \<in> B" and "a \<le> b" by blast

   107   from `b \<in> B` have "b \<le> Sup B" by (rule Sup_upper)

   108   with `a \<le> b` show "a \<le> Sup B" by auto

   109 qed

   111 definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where

   112   "INFI A f = \<Sqinter> (f ` A)"

   114 definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where

   115   "SUPR A f = \<Squnion> (f ` A)"

   117 end

   119 syntax

   120   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3INF _./ _)" [0, 10] 10)

   121   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3INF _:_./ _)" [0, 0, 10] 10)

   122   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3SUP _./ _)" [0, 10] 10)

   123   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3SUP _:_./ _)" [0, 0, 10] 10)

   125 syntax (xsymbols)

   126   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)

   127   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)

   128   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)

   129   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)

   131 translations

   132   "INF x y. B"   == "INF x. INF y. B"

   133   "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"

   134   "INF x. B"     == "INF x:CONST UNIV. B"

   135   "INF x:A. B"   == "CONST INFI A (%x. B)"

   136   "SUP x y. B"   == "SUP x. SUP y. B"

   137   "SUP x. B"     == "CONST SUPR CONST UNIV (%x. B)"

   138   "SUP x. B"     == "SUP x:CONST UNIV. B"

   139   "SUP x:A. B"   == "CONST SUPR A (%x. B)"

   141 print_translation {*

   142   [Syntax.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},

   143     Syntax.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]

   144 *} -- {* to avoid eta-contraction of body *}

   146 context complete_lattice

   147 begin

   149 lemma le_SUPI: "i : A \<Longrightarrow> M i \<sqsubseteq> (SUP i:A. M i)"

   150   by (auto simp add: SUPR_def intro: Sup_upper)

   152 lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (SUP i:A. M i) \<sqsubseteq> u"

   153   by (auto simp add: SUPR_def intro: Sup_least)

   155 lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<sqsubseteq> M i"

   156   by (auto simp add: INFI_def intro: Inf_lower)

   158 lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (INF i:A. M i)"

   159   by (auto simp add: INFI_def intro: Inf_greatest)

   161 lemma SUP_le_iff: "(SUP i:A. M i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. M i \<sqsubseteq> u)"

   162   unfolding SUPR_def by (auto simp add: Sup_le_iff)

   164 lemma le_INF_iff: "u \<sqsubseteq> (INF i:A. M i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> M i)"

   165   unfolding INFI_def by (auto simp add: le_Inf_iff)

   167 lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"

   168   by (auto intro: antisym INF_leI le_INFI)

   170 lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"

   171   by (auto intro: antisym SUP_leI le_SUPI)

   173 lemma INF_mono:

   174   "(\<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)"

   175   by (force intro!: Inf_mono simp: INFI_def)

   177 lemma SUP_mono:

   178   "(\<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)"

   179   by (force intro!: Sup_mono simp: SUPR_def)

   181 lemma INF_subset:  "A \<subseteq> B \<Longrightarrow> INFI B f \<le> INFI A f"

   182   by (intro INF_mono) auto

   184 lemma SUP_subset:  "A \<subseteq> B \<Longrightarrow> SUPR A f \<le> SUPR B f"

   185   by (intro SUP_mono) auto

   187 lemma INF_commute: "(INF i:A. INF j:B. f i j) = (INF j:B. INF i:A. f i j)"

   188   by (iprover intro: INF_leI le_INFI order_trans antisym)

   190 lemma SUP_commute: "(SUP i:A. SUP j:B. f i j) = (SUP j:B. SUP i:A. f i j)"

   191   by (iprover intro: SUP_leI le_SUPI order_trans antisym)

   193 end

   195 lemma Inf_less_iff:

   196   fixes a :: "'a\<Colon>{complete_lattice,linorder}"

   197   shows "Inf S < a \<longleftrightarrow> (\<exists>x\<in>S. x < a)"

   198   unfolding not_le[symmetric] le_Inf_iff by auto

   200 lemma less_Sup_iff:

   201   fixes a :: "'a\<Colon>{complete_lattice,linorder}"

   202   shows "a < Sup S \<longleftrightarrow> (\<exists>x\<in>S. a < x)"

   203   unfolding not_le[symmetric] Sup_le_iff by auto

   205 lemma INF_less_iff:

   206   fixes a :: "'a::{complete_lattice,linorder}"

   207   shows "(INF i:A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)"

   208   unfolding INFI_def Inf_less_iff by auto

   210 lemma less_SUP_iff:

   211   fixes a :: "'a::{complete_lattice,linorder}"

   212   shows "a < (SUP i:A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < f x)"

   213   unfolding SUPR_def less_Sup_iff by auto

   215 subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}

   217 instantiation bool :: complete_lattice

   218 begin

   220 definition

   221   "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"

   223 definition

   224   "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"

   226 instance proof

   227 qed (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)

   229 end

   231 lemma INFI_bool_eq [simp]:

   232   "INFI = Ball"

   233 proof (rule ext)+

   234   fix A :: "'a set"

   235   fix P :: "'a \<Rightarrow> bool"

   236   show "(INF x:A. P x) \<longleftrightarrow> (\<forall>x \<in> A. P x)"

   237     by (auto simp add: Ball_def INFI_def Inf_bool_def)

   238 qed

   240 lemma SUPR_bool_eq [simp]:

   241   "SUPR = Bex"

   242 proof (rule ext)+

   243   fix A :: "'a set"

   244   fix P :: "'a \<Rightarrow> bool"

   245   show "(SUP x:A. P x) \<longleftrightarrow> (\<exists>x \<in> A. P x)"

   246     by (auto simp add: Bex_def SUPR_def Sup_bool_def)

   247 qed

   249 instantiation "fun" :: (type, complete_lattice) complete_lattice

   250 begin

   252 definition

   253   "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"

   255 lemma Inf_apply:

   256   "(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}"

   257   by (simp add: Inf_fun_def)

   259 definition

   260   "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"

   262 lemma Sup_apply:

   263   "(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}"

   264   by (simp add: Sup_fun_def)

   266 instance proof

   267 qed (auto simp add: le_fun_def Inf_apply Sup_apply

   268   intro: Inf_lower Sup_upper Inf_greatest Sup_least)

   270 end

   272 lemma INFI_apply:

   273   "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"

   274   by (auto intro: arg_cong [of _ _ Inf] simp add: INFI_def Inf_apply)

   276 lemma SUPR_apply:

   277   "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"

   278   by (auto intro: arg_cong [of _ _ Sup] simp add: SUPR_def Sup_apply)

   281 subsection {* Inter *}

   283 abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where

   284   "Inter S \<equiv> \<Sqinter>S"

   286 notation (xsymbols)

   287   Inter  ("\<Inter>_" [90] 90)

   289 lemma Inter_eq:

   290   "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"

   291 proof (rule set_eqI)

   292   fix x

   293   have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"

   294     by auto

   295   then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"

   296     by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)

   297 qed

   299 lemma Inter_iff [simp,no_atp]: "(A : Inter C) = (ALL X:C. A:X)"

   300   by (unfold Inter_eq) blast

   302 lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"

   303   by (simp add: Inter_eq)

   305 text {*

   306   \medskip A ``destruct'' rule -- every @{term X} in @{term C}

   307   contains @{term A} as an element, but @{prop "A:X"} can hold when

   308   @{prop "X:C"} does not!  This rule is analogous to @{text spec}.

   309 *}

   311 lemma InterD [elim, Pure.elim]: "A : Inter C ==> X:C ==> A:X"

   312   by auto

   314 lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"

   315   -- {* ``Classical'' elimination rule -- does not require proving

   316     @{prop "X:C"}. *}

   317   by (unfold Inter_eq) blast

   319 lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"

   320   by blast

   322 lemma Inter_subset:

   323   "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"

   324   by blast

   326 lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"

   327   by (iprover intro: InterI subsetI dest: subsetD)

   329 lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"

   330   by blast

   332 lemma Inter_empty [simp]: "\<Inter>{} = UNIV"

   333   by (fact Inf_empty)

   335 lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"

   336   by blast

   338 lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"

   339   by blast

   341 lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"

   342   by blast

   344 lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"

   345   by blast

   347 lemma Inter_UNIV_conv [simp,no_atp]:

   348   "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"

   349   "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"

   350   by blast+

   352 lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"

   353   by blast

   356 subsection {* Intersections of families *}

   358 abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where

   359   "INTER \<equiv> INFI"

   361 syntax

   362   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)

   363   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)

   365 syntax (xsymbols)

   366   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)

   367   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)

   369 syntax (latex output)

   370   "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)

   371   "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)

   373 translations

   374   "INT x y. B"  == "INT x. INT y. B"

   375   "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"

   376   "INT x. B"    == "INT x:CONST UNIV. B"

   377   "INT x:A. B"  == "CONST INTER A (%x. B)"

   379 print_translation {*

   380   [Syntax.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]

   381 *} -- {* to avoid eta-contraction of body *}

   383 lemma INTER_eq_Inter_image:

   384   "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"

   385   by (fact INFI_def)

   387 lemma Inter_def:

   388   "\<Inter>S = (\<Inter>x\<in>S. x)"

   389   by (simp add: INTER_eq_Inter_image image_def)

   391 lemma INTER_def:

   392   "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"

   393   by (auto simp add: INTER_eq_Inter_image Inter_eq)

   395 lemma Inter_image_eq [simp]:

   396   "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"

   397   by (rule sym) (fact INTER_eq_Inter_image)

   399 lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"

   400   by (unfold INTER_def) blast

   402 lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"

   403   by (unfold INTER_def) blast

   405 lemma INT_D [elim, Pure.elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"

   406   by auto

   408 lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"

   409   -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}

   410   by (unfold INTER_def) blast

   412 lemma INT_cong [cong]:

   413     "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"

   414   by (simp add: INTER_def)

   416 lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"

   417   by blast

   419 lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"

   420   by blast

   422 lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"

   423   by (fact INF_leI)

   425 lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"

   426   by (fact le_INFI)

   428 lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"

   429   by blast

   431 lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"

   432   by blast

   434 lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"

   435   by (fact le_INF_iff)

   437 lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"

   438   by blast

   440 lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"

   441   by blast

   443 lemma INT_insert_distrib:

   444     "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"

   445   by blast

   447 lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"

   448   by auto

   450 lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"

   451   -- {* Look: it has an \emph{existential} quantifier *}

   452   by blast

   454 lemma INTER_UNIV_conv[simp]:

   455  "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"

   456  "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"

   457 by blast+

   459 lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"

   460   by (auto intro: bool_induct)

   462 lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"

   463   by blast

   465 lemma INT_anti_mono:

   466   "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>

   467     (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"

   468   -- {* The last inclusion is POSITIVE! *}

   469   by (blast dest: subsetD)

   471 lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"

   472   by blast

   475 subsection {* Union *}

   477 abbreviation Union :: "'a set set \<Rightarrow> 'a set" where

   478   "Union S \<equiv> \<Squnion>S"

   480 notation (xsymbols)

   481   Union  ("\<Union>_" [90] 90)

   483 lemma Union_eq:

   484   "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"

   485 proof (rule set_eqI)

   486   fix x

   487   have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"

   488     by auto

   489   then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"

   490     by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)

   491 qed

   493 lemma Union_iff [simp, no_atp]:

   494   "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"

   495   by (unfold Union_eq) blast

   497 lemma UnionI [intro]:

   498   "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"

   499   -- {* The order of the premises presupposes that @{term C} is rigid;

   500     @{term A} may be flexible. *}

   501   by auto

   503 lemma UnionE [elim!]:

   504   "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A\<in>X \<Longrightarrow> X\<in>C \<Longrightarrow> R) \<Longrightarrow> R"

   505   by auto

   507 lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"

   508   by (iprover intro: subsetI UnionI)

   510 lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"

   511   by (iprover intro: subsetI elim: UnionE dest: subsetD)

   513 lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"

   514   by blast

   516 lemma Union_empty [simp]: "Union({}) = {}"

   517   by blast

   519 lemma Union_UNIV [simp]: "Union UNIV = UNIV"

   520   by blast

   522 lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"

   523   by blast

   525 lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"

   526   by blast

   528 lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"

   529   by blast

   531 lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"

   532   by blast

   534 lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"

   535   by blast

   537 lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"

   538   by blast

   540 lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"

   541   by blast

   543 lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"

   544   by blast

   546 lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"

   547   by blast

   550 subsection {* Unions of families *}

   552 abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where

   553   "UNION \<equiv> SUPR"

   555 syntax

   556   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)

   557   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)

   559 syntax (xsymbols)

   560   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)

   561   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)

   563 syntax (latex output)

   564   "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)

   565   "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)

   567 translations

   568   "UN x y. B"   == "UN x. UN y. B"

   569   "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"

   570   "UN x. B"     == "UN x:CONST UNIV. B"

   571   "UN x:A. B"   == "CONST UNION A (%x. B)"

   573 text {*

   574   Note the difference between ordinary xsymbol syntax of indexed

   575   unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})

   576   and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The

   577   former does not make the index expression a subscript of the

   578   union/intersection symbol because this leads to problems with nested

   579   subscripts in Proof General.

   580 *}

   582 print_translation {*

   583   [Syntax.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]

   584 *} -- {* to avoid eta-contraction of body *}

   586 lemma UNION_eq_Union_image:

   587   "(\<Union>x\<in>A. B x) = \<Union>(B`A)"

   588   by (fact SUPR_def)

   590 lemma Union_def:

   591   "\<Union>S = (\<Union>x\<in>S. x)"

   592   by (simp add: UNION_eq_Union_image image_def)

   594 lemma UNION_def [no_atp]:

   595   "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"

   596   by (auto simp add: UNION_eq_Union_image Union_eq)

   598 lemma Union_image_eq [simp]:

   599   "\<Union>(B`A) = (\<Union>x\<in>A. B x)"

   600   by (rule sym) (fact UNION_eq_Union_image)

   602 lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"

   603   by (unfold UNION_def) blast

   605 lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"

   606   -- {* The order of the premises presupposes that @{term A} is rigid;

   607     @{term b} may be flexible. *}

   608   by auto

   610 lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"

   611   by (unfold UNION_def) blast

   613 lemma UN_cong [cong]:

   614     "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"

   615   by (simp add: UNION_def)

   617 lemma strong_UN_cong:

   618     "A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"

   619   by (simp add: UNION_def simp_implies_def)

   621 lemma image_eq_UN: "f`A = (UN x:A. {f x})"

   622   by blast

   624 lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"

   625   by (fact le_SUPI)

   627 lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"

   628   by (iprover intro: subsetI elim: UN_E dest: subsetD)

   630 lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"

   631   by blast

   633 lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"

   634   by blast

   636 lemma UN_empty [simp,no_atp]: "(\<Union>x\<in>{}. B x) = {}"

   637   by blast

   639 lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"

   640   by blast

   642 lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"

   643   by blast

   645 lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"

   646   by auto

   648 lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"

   649   by blast

   651 lemma UN_Un[simp]: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"

   652   by blast

   654 lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)"

   655   by blast

   657 lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"

   658   by (fact SUP_le_iff)

   660 lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"

   661   by blast

   663 lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"

   664   by auto

   666 lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"

   667   by blast

   669 lemma UNION_empty_conv[simp]:

   670   "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"

   671   "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"

   672 by blast+

   674 lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"

   675   by blast

   677 lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"

   678   by blast

   680 lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"

   681   by blast

   683 lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"

   684   by (auto simp add: split_if_mem2)

   686 lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"

   687   by (auto intro: bool_contrapos)

   689 lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"

   690   by blast

   692 lemma UN_mono:

   693   "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>

   694     (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"

   695   by (blast dest: subsetD)

   697 lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"

   698   by blast

   700 lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"

   701   by blast

   703 lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"

   704   -- {* NOT suitable for rewriting *}

   705   by blast

   707 lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"

   708 by blast

   711 subsection {* Distributive laws *}

   713 lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"

   714   by blast

   716 lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"

   717   by blast

   719 lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"

   720   -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}

   721   -- {* Union of a family of unions *}

   722   by blast

   724 lemma UN_Un_distrib: "(\<Union>i\<in>I. A i \<union> B i) = (\<Union>i\<in>I. A i) \<union> (\<Union>i\<in>I. B i)"

   725   -- {* Equivalent version *}

   726   by blast

   728 lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"

   729   by blast

   731 lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"

   732   by blast

   734 lemma INT_Int_distrib: "(\<Inter>i\<in>I. A i \<inter> B i) = (\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i)"

   735   -- {* Equivalent version *}

   736   by blast

   738 lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"

   739   -- {* Halmos, Naive Set Theory, page 35. *}

   740   by blast

   742 lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"

   743   by blast

   745 lemma Int_UN_distrib2: "(\<Union>i\<in>I. A i) \<inter> (\<Union>j\<in>J. B j) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)"

   746   by blast

   748 lemma Un_INT_distrib2: "(\<Inter>i\<in>I. A i) \<union> (\<Inter>j\<in>J. B j) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A i \<union> B j)"

   749   by blast

   752 subsection {* Complement *}

   754 lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"

   755   by blast

   757 lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"

   758   by blast

   761 subsection {* Miniscoping and maxiscoping *}

   763 text {* \medskip Miniscoping: pushing in quantifiers and big Unions

   764            and Intersections. *}

   766 lemma UN_simps [simp]:

   767   "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"

   768   "!!A B C. (UN x:C. A x Un B)   = ((if C={} then {} else (UN x:C. A x) Un B))"

   769   "!!A B C. (UN x:C. A Un B x)   = ((if C={} then {} else A Un (UN x:C. B x)))"

   770   "!!A B C. (UN x:C. A x Int B)  = ((UN x:C. A x) Int B)"

   771   "!!A B C. (UN x:C. A Int B x)  = (A Int (UN x:C. B x))"

   772   "!!A B C. (UN x:C. A x - B)    = ((UN x:C. A x) - B)"

   773   "!!A B C. (UN x:C. A - B x)    = (A - (INT x:C. B x))"

   774   "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"

   775   "!!A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"

   776   "!!A B f. (UN x:f`A. B x)     = (UN a:A. B (f a))"

   777   by auto

   779 lemma INT_simps [simp]:

   780   "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"

   781   "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"

   782   "!!A B C. (INT x:C. A x - B)   = (if C={} then UNIV else (INT x:C. A x) - B)"

   783   "!!A B C. (INT x:C. A - B x)   = (if C={} then UNIV else A - (UN x:C. B x))"

   784   "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"

   785   "!!A B C. (INT x:C. A x Un B)  = ((INT x:C. A x) Un B)"

   786   "!!A B C. (INT x:C. A Un B x)  = (A Un (INT x:C. B x))"

   787   "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"

   788   "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"

   789   "!!A B f. (INT x:f`A. B x)    = (INT a:A. B (f a))"

   790   by auto

   792 lemma ball_simps [simp,no_atp]:

   793   "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"

   794   "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"

   795   "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"

   796   "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"

   797   "!!P. (ALL x:{}. P x) = True"

   798   "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"

   799   "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"

   800   "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"

   801   "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"

   802   "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"

   803   "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"

   804   "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"

   805   by auto

   807 lemma bex_simps [simp,no_atp]:

   808   "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"

   809   "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"

   810   "!!P. (EX x:{}. P x) = False"

   811   "!!P. (EX x:UNIV. P x) = (EX x. P x)"

   812   "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"

   813   "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"

   814   "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"

   815   "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"

   816   "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"

   817   "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"

   818   by auto

   820 lemma ball_conj_distrib:

   821   "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"

   822   by blast

   824 lemma bex_disj_distrib:

   825   "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"

   826   by blast

   829 text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}

   831 lemma UN_extend_simps:

   832   "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"

   833   "!!A B C. (UN x:C. A x) Un B    = (if C={} then B else (UN x:C. A x Un B))"

   834   "!!A B C. A Un (UN x:C. B x)   = (if C={} then A else (UN x:C. A Un B x))"

   835   "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"

   836   "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"

   837   "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"

   838   "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"

   839   "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"

   840   "!!A B C. (UN  x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"

   841   "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"

   842   by auto

   844 lemma INT_extend_simps:

   845   "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"

   846   "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"

   847   "!!A B C. (INT x:C. A x) - B   = (if C={} then UNIV-B else (INT x:C. A x - B))"

   848   "!!A B C. A - (UN x:C. B x)   = (if C={} then A else (INT x:C. A - B x))"

   849   "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"

   850   "!!A B C. ((INT x:C. A x) Un B)  = (INT x:C. A x Un B)"

   851   "!!A B C. A Un (INT x:C. B x)  = (INT x:C. A Un B x)"

   852   "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"

   853   "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"

   854   "!!A B f. (INT a:A. B (f a))    = (INT x:f`A. B x)"

   855   by auto

   858 no_notation

   859   less_eq  (infix "\<sqsubseteq>" 50) and

   860   less (infix "\<sqsubset>" 50) and

   861   bot ("\<bottom>") and

   862   top ("\<top>") and

   863   inf  (infixl "\<sqinter>" 70) and

   864   sup  (infixl "\<squnion>" 65) and

   865   Inf  ("\<Sqinter>_" [900] 900) and

   866   Sup  ("\<Squnion>_" [900] 900)

   868 no_syntax (xsymbols)

   869   "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)

   870   "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)

   871   "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)

   872   "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)

   874 lemmas mem_simps =

   875   insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff

   876   mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff

   877   -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}

   879 end

author	haftmann
	Wed, 08 Dec 2010 15:05:46 +0100
changeset 41330	9ff94e7cc3b3
parent 41328	294956ff285b
child 42842	a54e8e95fe96
permissions	-rw-r--r--