(* Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
header {* Complete lattices, with special focus on sets *}
theory Complete_Lattice
imports Set
begin
notation
less_eq (infix "\<sqsubseteq>" 50) and
less (infix "\<sqsubset>" 50) and
inf (infixl "\<sqinter>" 70) and
sup (infixl "\<squnion>" 65) and
top ("\<top>") and
bot ("\<bottom>")
subsection {* Syntactic infimum and supremum operations *}
class Inf =
fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
class Sup =
fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
subsection {* Abstract complete lattices *}
class complete_lattice = bounded_lattice + Inf + Sup +
assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
begin
lemma dual_complete_lattice:
"class.complete_lattice Sup Inf (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom>"
by (auto intro!: class.complete_lattice.intro dual_bounded_lattice)
(unfold_locales, (fact bot_least top_greatest
Sup_upper Sup_least Inf_lower Inf_greatest)+)
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}"
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
lemma Inf_empty [simp]:
"\<Sqinter>{} = \<top>"
by (auto intro: antisym Inf_greatest)
lemma Sup_empty [simp]:
"\<Squnion>{} = \<bottom>"
by (auto intro: antisym Sup_least)
lemma Inf_UNIV [simp]:
"\<Sqinter>UNIV = \<bottom>"
by (simp add: Sup_Inf Sup_empty [symmetric])
lemma Sup_UNIV [simp]:
"\<Squnion>UNIV = \<top>"
by (simp add: Inf_Sup Inf_empty [symmetric])
lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
lemma Inf_singleton [simp]:
"\<Sqinter>{a} = a"
by (auto intro: antisym Inf_lower Inf_greatest)
lemma Sup_singleton [simp]:
"\<Squnion>{a} = a"
by (auto intro: antisym Sup_upper Sup_least)
lemma Inf_binary:
"\<Sqinter>{a, b} = a \<sqinter> b"
by (simp add: Inf_empty Inf_insert)
lemma Sup_binary:
"\<Squnion>{a, b} = a \<squnion> b"
by (simp add: Sup_empty Sup_insert)
lemma le_Inf_iff: "b \<sqsubseteq> Inf A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
by (auto intro: Inf_greatest dest: Inf_lower)
lemma Sup_le_iff: "Sup A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
by (auto intro: Sup_least dest: Sup_upper)
lemma Inf_mono:
assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<le> b"
shows "Inf A \<le> Inf B"
proof (rule Inf_greatest)
fix b assume "b \<in> B"
with assms obtain a where "a \<in> A" and "a \<le> b" by blast
from `a \<in> A` have "Inf A \<le> a" by (rule Inf_lower)
with `a \<le> b` show "Inf A \<le> b" by auto
qed
lemma Sup_mono:
assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<le> b"
shows "Sup A \<le> Sup B"
proof (rule Sup_least)
fix a assume "a \<in> A"
with assms obtain b where "b \<in> B" and "a \<le> b" by blast
from `b \<in> B` have "b \<le> Sup B" by (rule Sup_upper)
with `a \<le> b` show "a \<le> Sup B" by auto
qed
definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
"INFI A f = \<Sqinter> (f ` A)"
definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
"SUPR A f = \<Squnion> (f ` A)"
end
syntax
"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _./ _)" [0, 10] 10)
"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _:_./ _)" [0, 0, 10] 10)
"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _./ _)" [0, 10] 10)
"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _:_./ _)" [0, 0, 10] 10)
syntax (xsymbols)
"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10)
"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
translations
"INF x y. B" == "INF x. INF y. B"
"INF x. B" == "CONST INFI CONST UNIV (%x. B)"
"INF x. B" == "INF x:CONST UNIV. B"
"INF x:A. B" == "CONST INFI A (%x. B)"
"SUP x y. B" == "SUP x. SUP y. B"
"SUP x. B" == "CONST SUPR CONST UNIV (%x. B)"
"SUP x. B" == "SUP x:CONST UNIV. B"
"SUP x:A. B" == "CONST SUPR A (%x. B)"
print_translation {*
[Syntax.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
Syntax.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
*} -- {* to avoid eta-contraction of body *}
context complete_lattice
begin
lemma le_SUPI: "i : A \<Longrightarrow> M i \<sqsubseteq> (SUP i:A. M i)"
by (auto simp add: SUPR_def intro: Sup_upper)
lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (SUP i:A. M i) \<sqsubseteq> u"
by (auto simp add: SUPR_def intro: Sup_least)
lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<sqsubseteq> M i"
by (auto simp add: INFI_def intro: Inf_lower)
lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (INF i:A. M i)"
by (auto simp add: INFI_def intro: Inf_greatest)
lemma SUP_le_iff: "(SUP i:A. M i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. M i \<sqsubseteq> u)"
unfolding SUPR_def by (auto simp add: Sup_le_iff)
lemma le_INF_iff: "u \<sqsubseteq> (INF i:A. M i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> M i)"
unfolding INFI_def by (auto simp add: le_Inf_iff)
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
by (auto intro: antisym INF_leI le_INFI)
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
by (auto intro: antisym SUP_leI le_SUPI)
lemma INF_mono:
"(\<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)"
by (force intro!: Inf_mono simp: INFI_def)
lemma SUP_mono:
"(\<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)"
by (force intro!: Sup_mono simp: SUPR_def)
lemma INF_subset: "A \<subseteq> B \<Longrightarrow> INFI B f \<le> INFI A f"
by (intro INF_mono) auto
lemma SUP_subset: "A \<subseteq> B \<Longrightarrow> SUPR A f \<le> SUPR B f"
by (intro SUP_mono) auto
lemma INF_commute: "(INF i:A. INF j:B. f i j) = (INF j:B. INF i:A. f i j)"
by (iprover intro: INF_leI le_INFI order_trans antisym)
lemma SUP_commute: "(SUP i:A. SUP j:B. f i j) = (SUP j:B. SUP i:A. f i j)"
by (iprover intro: SUP_leI le_SUPI order_trans antisym)
end
lemma Inf_less_iff:
fixes a :: "'a\<Colon>{complete_lattice,linorder}"
shows "Inf S < a \<longleftrightarrow> (\<exists>x\<in>S. x < a)"
unfolding not_le[symmetric] le_Inf_iff by auto
lemma less_Sup_iff:
fixes a :: "'a\<Colon>{complete_lattice,linorder}"
shows "a < Sup S \<longleftrightarrow> (\<exists>x\<in>S. a < x)"
unfolding not_le[symmetric] Sup_le_iff by auto
lemma INF_less_iff:
fixes a :: "'a::{complete_lattice,linorder}"
shows "(INF i:A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)"
unfolding INFI_def Inf_less_iff by auto
lemma less_SUP_iff:
fixes a :: "'a::{complete_lattice,linorder}"
shows "a < (SUP i:A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < f x)"
unfolding SUPR_def less_Sup_iff by auto
subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
instantiation bool :: complete_lattice
begin
definition
"\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
definition
"\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
instance proof
qed (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
end
lemma INFI_bool_eq [simp]:
"INFI = Ball"
proof (rule ext)+
fix A :: "'a set"
fix P :: "'a \<Rightarrow> bool"
show "(INF x:A. P x) \<longleftrightarrow> (\<forall>x \<in> A. P x)"
by (auto simp add: Ball_def INFI_def Inf_bool_def)
qed
lemma SUPR_bool_eq [simp]:
"SUPR = Bex"
proof (rule ext)+
fix A :: "'a set"
fix P :: "'a \<Rightarrow> bool"
show "(SUP x:A. P x) \<longleftrightarrow> (\<exists>x \<in> A. P x)"
by (auto simp add: Bex_def SUPR_def Sup_bool_def)
qed
instantiation "fun" :: (type, complete_lattice) complete_lattice
begin
definition
"\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
lemma Inf_apply:
"(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}"
by (simp add: Inf_fun_def)
definition
"\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
lemma Sup_apply:
"(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}"
by (simp add: Sup_fun_def)
instance proof
qed (auto simp add: le_fun_def Inf_apply Sup_apply
intro: Inf_lower Sup_upper Inf_greatest Sup_least)
end
lemma INFI_apply:
"(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
by (auto intro: arg_cong [of _ _ Inf] simp add: INFI_def Inf_apply)
lemma SUPR_apply:
"(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
by (auto intro: arg_cong [of _ _ Sup] simp add: SUPR_def Sup_apply)
subsection {* Inter *}
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
"Inter S \<equiv> \<Sqinter>S"
notation (xsymbols)
Inter ("\<Inter>_" [90] 90)
lemma Inter_eq:
"\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
proof (rule set_eqI)
fix x
have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
by auto
then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
qed
lemma Inter_iff [simp,no_atp]: "(A : Inter C) = (ALL X:C. A:X)"
by (unfold Inter_eq) blast
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
by (simp add: Inter_eq)
text {*
\medskip A ``destruct'' rule -- every @{term X} in @{term C}
contains @{term A} as an element, but @{prop "A:X"} can hold when
@{prop "X:C"} does not! This rule is analogous to @{text spec}.
*}
lemma InterD [elim, Pure.elim]: "A : Inter C ==> X:C ==> A:X"
by auto
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
-- {* ``Classical'' elimination rule -- does not require proving
@{prop "X:C"}. *}
by (unfold Inter_eq) blast
lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
by blast
lemma Inter_subset:
"[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
by blast
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
by (iprover intro: InterI subsetI dest: subsetD)
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
by blast
lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
by (fact Inf_empty)
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
by blast
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
by blast
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
by blast
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
by blast
lemma Inter_UNIV_conv [simp,no_atp]:
"(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
"(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
by blast+
lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"
by blast
subsection {* Intersections of families *}
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
"INTER \<equiv> INFI"
syntax
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10)
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 0, 10] 10)
syntax (xsymbols)
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10)
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
syntax (latex output)
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
translations
"INT x y. B" == "INT x. INT y. B"
"INT x. B" == "CONST INTER CONST UNIV (%x. B)"
"INT x. B" == "INT x:CONST UNIV. B"
"INT x:A. B" == "CONST INTER A (%x. B)"
print_translation {*
[Syntax.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
*} -- {* to avoid eta-contraction of body *}
lemma INTER_eq_Inter_image:
"(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
by (fact INFI_def)
lemma Inter_def:
"\<Inter>S = (\<Inter>x\<in>S. x)"
by (simp add: INTER_eq_Inter_image image_def)
lemma INTER_def:
"(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
by (auto simp add: INTER_eq_Inter_image Inter_eq)
lemma Inter_image_eq [simp]:
"\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
by (rule sym) (fact INTER_eq_Inter_image)
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
by (unfold INTER_def) blast
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
by (unfold INTER_def) blast
lemma INT_D [elim, Pure.elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
by auto
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
-- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
by (unfold INTER_def) blast
lemma INT_cong [cong]:
"A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
by (simp add: INTER_def)
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
by blast
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
by blast
lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
by (fact INF_leI)
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
by (fact le_INFI)
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
by blast
lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
by blast
lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
by (fact le_INF_iff)
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
by blast
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
by blast
lemma INT_insert_distrib:
"u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
by blast
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
by auto
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
-- {* Look: it has an \emph{existential} quantifier *}
by blast
lemma INTER_UNIV_conv[simp]:
"(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
"((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
by blast+
lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
by (auto intro: bool_induct)
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
by blast
lemma INT_anti_mono:
"B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
(\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
-- {* The last inclusion is POSITIVE! *}
by (blast dest: subsetD)
lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
by blast
subsection {* Union *}
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
"Union S \<equiv> \<Squnion>S"
notation (xsymbols)
Union ("\<Union>_" [90] 90)
lemma Union_eq:
"\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
proof (rule set_eqI)
fix x
have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
by auto
then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
qed
lemma Union_iff [simp, no_atp]:
"A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
by (unfold Union_eq) blast
lemma UnionI [intro]:
"X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
-- {* The order of the premises presupposes that @{term C} is rigid;
@{term A} may be flexible. *}
by auto
lemma UnionE [elim!]:
"A \<in> \<Union>C \<Longrightarrow> (\<And>X. A\<in>X \<Longrightarrow> X\<in>C \<Longrightarrow> R) \<Longrightarrow> R"
by auto
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
by (iprover intro: subsetI UnionI)
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
by (iprover intro: subsetI elim: UnionE dest: subsetD)
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
by blast
lemma Union_empty [simp]: "Union({}) = {}"
by blast
lemma Union_UNIV [simp]: "Union UNIV = UNIV"
by blast
lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
by blast
lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
by blast
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
by blast
lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
by blast
lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
by blast
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
by blast
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
by blast
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
by blast
lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"
by blast
subsection {* Unions of families *}
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
"UNION \<equiv> SUPR"
syntax
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" [0, 10] 10)
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" [0, 0, 10] 10)
syntax (xsymbols)
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10)
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
syntax (latex output)
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
translations
"UN x y. B" == "UN x. UN y. B"
"UN x. B" == "CONST UNION CONST UNIV (%x. B)"
"UN x. B" == "UN x:CONST UNIV. B"
"UN x:A. B" == "CONST UNION A (%x. B)"
text {*
Note the difference between ordinary xsymbol syntax of indexed
unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
former does not make the index expression a subscript of the
union/intersection symbol because this leads to problems with nested
subscripts in Proof General.
*}
print_translation {*
[Syntax.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
*} -- {* to avoid eta-contraction of body *}
lemma UNION_eq_Union_image:
"(\<Union>x\<in>A. B x) = \<Union>(B`A)"
by (fact SUPR_def)
lemma Union_def:
"\<Union>S = (\<Union>x\<in>S. x)"
by (simp add: UNION_eq_Union_image image_def)
lemma UNION_def [no_atp]:
"(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
by (auto simp add: UNION_eq_Union_image Union_eq)
lemma Union_image_eq [simp]:
"\<Union>(B`A) = (\<Union>x\<in>A. B x)"
by (rule sym) (fact UNION_eq_Union_image)
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
by (unfold UNION_def) blast
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
-- {* The order of the premises presupposes that @{term A} is rigid;
@{term b} may be flexible. *}
by auto
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
by (unfold UNION_def) blast
lemma UN_cong [cong]:
"A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
by (simp add: UNION_def)
lemma strong_UN_cong:
"A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
by (simp add: UNION_def simp_implies_def)
lemma image_eq_UN: "f`A = (UN x:A. {f x})"
by blast
lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
by (fact le_SUPI)
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
by (iprover intro: subsetI elim: UN_E dest: subsetD)
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
by blast
lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
by blast
lemma UN_empty [simp,no_atp]: "(\<Union>x\<in>{}. B x) = {}"
by blast
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
by blast
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
by blast
lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
by auto
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
by blast
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)"
by blast
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)"
by blast
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
by (fact SUP_le_iff)
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
by blast
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
by auto
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
by blast
lemma UNION_empty_conv[simp]:
"({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
"((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
by blast+
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
by blast
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
by blast
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
by blast
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
by (auto simp add: split_if_mem2)
lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
by (auto intro: bool_contrapos)
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
by blast
lemma UN_mono:
"A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
(\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
by (blast dest: subsetD)
lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"
by blast
lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"
by blast
lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"
-- {* NOT suitable for rewriting *}
by blast
lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
by blast
subsection {* Distributive laws *}
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
by blast
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
by blast
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"
-- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
-- {* Union of a family of unions *}
by blast
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)"
-- {* Equivalent version *}
by blast
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
by blast
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"
by blast
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)"
-- {* Equivalent version *}
by blast
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
-- {* Halmos, Naive Set Theory, page 35. *}
by blast
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
by blast
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)"
by blast
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)"
by blast
subsection {* Complement *}
lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
by blast
lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
by blast
subsection {* Miniscoping and maxiscoping *}
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
and Intersections. *}
lemma UN_simps [simp]:
"!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
"!!A B C. (UN x:C. A x Un B) = ((if C={} then {} else (UN x:C. A x) Un B))"
"!!A B C. (UN x:C. A Un B x) = ((if C={} then {} else A Un (UN x:C. B x)))"
"!!A B C. (UN x:C. A x Int B) = ((UN x:C. A x) Int B)"
"!!A B C. (UN x:C. A Int B x) = (A Int (UN x:C. B x))"
"!!A B C. (UN x:C. A x - B) = ((UN x:C. A x) - B)"
"!!A B C. (UN x:C. A - B x) = (A - (INT x:C. B x))"
"!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
"!!A B C. (UN z: UNION A B. C z) = (UN x:A. UN z: B(x). C z)"
"!!A B f. (UN x:f`A. B x) = (UN a:A. B (f a))"
by auto
lemma INT_simps [simp]:
"!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
"!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
"!!A B C. (INT x:C. A x - B) = (if C={} then UNIV else (INT x:C. A x) - B)"
"!!A B C. (INT x:C. A - B x) = (if C={} then UNIV else A - (UN x:C. B x))"
"!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
"!!A B C. (INT x:C. A x Un B) = ((INT x:C. A x) Un B)"
"!!A B C. (INT x:C. A Un B x) = (A Un (INT x:C. B x))"
"!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
"!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
"!!A B f. (INT x:f`A. B x) = (INT a:A. B (f a))"
by auto
lemma ball_simps [simp,no_atp]:
"!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
"!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
"!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
"!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
"!!P. (ALL x:{}. P x) = True"
"!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
"!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
"!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
"!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
"!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
"!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"
"!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
by auto
lemma bex_simps [simp,no_atp]:
"!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
"!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
"!!P. (EX x:{}. P x) = False"
"!!P. (EX x:UNIV. P x) = (EX x. P x)"
"!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
"!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
"!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
"!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
"!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"
"!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
by auto
lemma ball_conj_distrib:
"(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
by blast
lemma bex_disj_distrib:
"(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
by blast
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
lemma UN_extend_simps:
"!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
"!!A B C. (UN x:C. A x) Un B = (if C={} then B else (UN x:C. A x Un B))"
"!!A B C. A Un (UN x:C. B x) = (if C={} then A else (UN x:C. A Un B x))"
"!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
"!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
"!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
"!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
"!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
"!!A B C. (UN x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
"!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
by auto
lemma INT_extend_simps:
"!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
"!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
"!!A B C. (INT x:C. A x) - B = (if C={} then UNIV-B else (INT x:C. A x - B))"
"!!A B C. A - (UN x:C. B x) = (if C={} then A else (INT x:C. A - B x))"
"!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
"!!A B C. ((INT x:C. A x) Un B) = (INT x:C. A x Un B)"
"!!A B C. A Un (INT x:C. B x) = (INT x:C. A Un B x)"
"!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
"!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
"!!A B f. (INT a:A. B (f a)) = (INT x:f`A. B x)"
by auto
no_notation
less_eq (infix "\<sqsubseteq>" 50) and
less (infix "\<sqsubset>" 50) and
bot ("\<bottom>") and
top ("\<top>") and
inf (infixl "\<sqinter>" 70) and
sup (infixl "\<squnion>" 65) and
Inf ("\<Sqinter>_" [900] 900) and
Sup ("\<Squnion>_" [900] 900)
no_syntax (xsymbols)
"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10)
"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
lemmas mem_simps =
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
-- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
end