(* Title : Filter.thy
ID : $Id$
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Conversion to Isar and new proofs by Lawrence C Paulson, 2004
*)
header{*Filters and Ultrafilters*}
theory Filter
imports Zorn
begin
constdefs
is_Filter :: "['a set set,'a set] => bool"
"is_Filter F S == (F <= Pow(S) & S \<in> F & {} ~: F &
(\<forall>u \<in> F. \<forall>v \<in> F. u Int v \<in> F) &
(\<forall>u v. u \<in> F & u <= v & v <= S --> v \<in> F))"
Filter :: "'a set => 'a set set set"
"Filter S == {X. is_Filter X S}"
(* free filter does not contain any finite set *)
Freefilter :: "'a set => 'a set set set"
"Freefilter S == {X. X \<in> Filter S & (\<forall>x \<in> X. ~ finite x)}"
Ultrafilter :: "'a set => 'a set set set"
"Ultrafilter S == {X. X \<in> Filter S & (\<forall>A \<in> Pow(S). A \<in> X | S - A \<in> X)}"
FreeUltrafilter :: "'a set => 'a set set set"
"FreeUltrafilter S == {X. X \<in> Ultrafilter S & (\<forall>x \<in> X. ~ finite x)}"
(* A locale makes proof of Ultrafilter Theorem more modular *)
locale (open) UFT =
fixes frechet :: "'a set => 'a set set"
and superfrechet :: "'a set => 'a set set set"
assumes not_finite_UNIV: "~finite (UNIV :: 'a set)"
defines frechet_def:
"frechet S == {A. finite (S - A)}"
and superfrechet_def:
"superfrechet S == {G. G \<in> Filter S & frechet S <= G}"
(*------------------------------------------------------------------
Properties of Filters and Freefilters -
rules for intro, destruction etc.
------------------------------------------------------------------*)
lemma is_FilterD1: "is_Filter X S ==> X <= Pow(S)"
apply (simp add: is_Filter_def)
done
lemma is_FilterD2: "is_Filter X S ==> X ~= {}"
apply (auto simp add: is_Filter_def)
done
lemma is_FilterD3: "is_Filter X S ==> {} ~: X"
apply (simp add: is_Filter_def)
done
lemma mem_FiltersetI: "is_Filter X S ==> X \<in> Filter S"
apply (simp add: Filter_def)
done
lemma mem_FiltersetD: "X \<in> Filter S ==> is_Filter X S"
apply (simp add: Filter_def)
done
lemma Filter_empty_not_mem: "X \<in> Filter S ==> {} ~: X"
apply (erule mem_FiltersetD [THEN is_FilterD3])
done
lemmas Filter_empty_not_memE = Filter_empty_not_mem [THEN notE, standard]
lemma mem_FiltersetD1: "[| X \<in> Filter S; A \<in> X; B \<in> X |] ==> A Int B \<in> X"
apply (unfold Filter_def is_Filter_def)
apply blast
done
lemma mem_FiltersetD2: "[| X \<in> Filter S; A \<in> X; A <= B; B <= S|] ==> B \<in> X"
apply (unfold Filter_def is_Filter_def)
apply blast
done
lemma mem_FiltersetD3: "[| X \<in> Filter S; A \<in> X |] ==> A \<in> Pow S"
apply (unfold Filter_def is_Filter_def)
apply blast
done
lemma mem_FiltersetD4: "X \<in> Filter S ==> S \<in> X"
apply (unfold Filter_def is_Filter_def)
apply blast
done
lemma is_FilterI:
"[| X <= Pow(S);
S \<in> X;
X ~= {};
{} ~: X;
\<forall>u \<in> X. \<forall>v \<in> X. u Int v \<in> X;
\<forall>u v. u \<in> X & u<=v & v<=S --> v \<in> X
|] ==> is_Filter X S"
apply (unfold is_Filter_def)
apply blast
done
lemma mem_FiltersetI2: "[| X <= Pow(S);
S \<in> X;
X ~= {};
{} ~: X;
\<forall>u \<in> X. \<forall>v \<in> X. u Int v \<in> X;
\<forall>u v. u \<in> X & u<=v & v<=S --> v \<in> X
|] ==> X \<in> Filter S"
by (blast intro: mem_FiltersetI is_FilterI)
lemma is_FilterE_lemma:
"is_Filter X S ==> X <= Pow(S) &
S \<in> X &
X ~= {} &
{} ~: X &
(\<forall>u \<in> X. \<forall>v \<in> X. u Int v \<in> X) &
(\<forall>u v. u \<in> X & u <= v & v<=S --> v \<in> X)"
apply (unfold is_Filter_def)
apply fast
done
lemma memFiltersetE_lemma:
"X \<in> Filter S ==> X <= Pow(S) &
S \<in> X &
X ~= {} &
{} ~: X &
(\<forall>u \<in> X. \<forall>v \<in> X. u Int v \<in> X) &
(\<forall>u v. u \<in> X & u <= v & v<=S --> v \<in> X)"
by (erule mem_FiltersetD [THEN is_FilterE_lemma])
lemma Freefilter_Filter: "X \<in> Freefilter S ==> X \<in> Filter S"
apply (simp add: Filter_def Freefilter_def)
done
lemma mem_Freefilter_not_finite: "X \<in> Freefilter S ==> \<forall>y \<in> X. ~finite(y)"
apply (simp add: Freefilter_def)
done
lemma mem_FreefiltersetD1: "[| X \<in> Freefilter S; x \<in> X |] ==> ~ finite x"
apply (blast dest!: mem_Freefilter_not_finite)
done
lemmas mem_FreefiltersetE1 = mem_FreefiltersetD1 [THEN notE, standard]
lemma mem_FreefiltersetD2: "[| X \<in> Freefilter S; finite x|] ==> x ~: X"
apply (blast dest!: mem_Freefilter_not_finite)
done
lemma mem_FreefiltersetI1:
"[| X \<in> Filter S; \<forall>x. ~(x \<in> X & finite x) |] ==> X \<in> Freefilter S"
by (simp add: Freefilter_def)
lemma mem_FreefiltersetI2:
"[| X \<in> Filter S; \<forall>x. (x ~: X | ~ finite x) |] ==> X \<in> Freefilter S"
by (simp add: Freefilter_def)
lemma Filter_Int_not_empty: "[| X \<in> Filter S; A \<in> X; B \<in> X |] ==> A Int B ~= {}"
apply (frule_tac A = "A" and B = "B" in mem_FiltersetD1)
apply (auto dest!: Filter_empty_not_mem)
done
lemmas Filter_Int_not_emptyE = Filter_Int_not_empty [THEN notE, standard]
subsection{*Ultrafilters and Free Ultrafilters*}
lemma Ultrafilter_Filter: "X \<in> Ultrafilter S ==> X \<in> Filter S"
apply (simp add: Ultrafilter_def)
done
lemma mem_UltrafiltersetD2:
"X \<in> Ultrafilter S ==> \<forall>A \<in> Pow(S). A \<in> X | S - A \<in> X"
by (auto simp add: Ultrafilter_def)
lemma mem_UltrafiltersetD3:
"[|X \<in> Ultrafilter S; A <= S; A ~: X |] ==> S - A \<in> X"
by (auto simp add: Ultrafilter_def)
lemma mem_UltrafiltersetD4:
"[|X \<in> Ultrafilter S; A <= S; S - A ~: X |] ==> A \<in> X"
by (auto simp add: Ultrafilter_def)
lemma mem_UltrafiltersetI:
"[| X \<in> Filter S;
\<forall>A \<in> Pow(S). A \<in> X | S - A \<in> X |] ==> X \<in> Ultrafilter S"
by (simp add: Ultrafilter_def)
lemma FreeUltrafilter_Ultrafilter:
"X \<in> FreeUltrafilter S ==> X \<in> Ultrafilter S"
by (auto simp add: Ultrafilter_def FreeUltrafilter_def)
lemma mem_FreeUltrafilter_not_finite:
"X \<in> FreeUltrafilter S ==> \<forall>y \<in> X. ~finite(y)"
by (simp add: FreeUltrafilter_def)
lemma mem_FreeUltrafiltersetD1: "[| X \<in> FreeUltrafilter S; x \<in> X |] ==> ~ finite x"
apply (blast dest!: mem_FreeUltrafilter_not_finite)
done
lemmas mem_FreeUltrafiltersetE1 = mem_FreeUltrafiltersetD1 [THEN notE, standard]
lemma mem_FreeUltrafiltersetD2: "[| X \<in> FreeUltrafilter S; finite x|] ==> x ~: X"
apply (blast dest!: mem_FreeUltrafilter_not_finite)
done
lemma mem_FreeUltrafiltersetI1:
"[| X \<in> Ultrafilter S;
\<forall>x. ~(x \<in> X & finite x) |] ==> X \<in> FreeUltrafilter S"
by (simp add: FreeUltrafilter_def)
lemma mem_FreeUltrafiltersetI2:
"[| X \<in> Ultrafilter S;
\<forall>x. (x ~: X | ~ finite x) |] ==> X \<in> FreeUltrafilter S"
by (simp add: FreeUltrafilter_def)
lemma FreeUltrafilter_iff:
"(X \<in> FreeUltrafilter S) = (X \<in> Freefilter S & (\<forall>x \<in> Pow(S). x \<in> X | S - x \<in> X))"
by (auto simp add: FreeUltrafilter_def Freefilter_def Ultrafilter_def)
(*-------------------------------------------------------------------
A Filter F on S is an ultrafilter iff it is a maximal filter
i.e. whenever G is a filter on I and F <= F then F = G
--------------------------------------------------------------------*)
(*---------------------------------------------------------------------
lemmas that shows existence of an extension to what was assumed to
be a maximal filter. Will be used to derive contradiction in proof of
property of ultrafilter
---------------------------------------------------------------------*)
lemma lemma_set_extend: "[| F ~= {}; A <= S |] ==> \<exists>x. x \<in> {X. X <= S & (\<exists>f \<in> F. A Int f <= X)}"
apply blast
done
lemma lemma_set_not_empty: "a \<in> X ==> X ~= {}"
apply (safe)
done
lemma lemma_empty_Int_subset_Compl: "x Int F <= {} ==> F <= - x"
apply blast
done
lemma mem_Filterset_disjI:
"[| F \<in> Filter S; A ~: F; A <= S|]
==> \<forall>B. B ~: F | ~ B <= A"
apply (unfold Filter_def is_Filter_def)
apply blast
done
lemma Ultrafilter_max_Filter: "F \<in> Ultrafilter S ==>
(F \<in> Filter S & (\<forall>G \<in> Filter S. F <= G --> F = G))"
apply (auto simp add: Ultrafilter_def)
apply (drule_tac x = "x" in bspec)
apply (erule mem_FiltersetD3 , assumption)
apply (safe)
apply (drule subsetD , assumption)
apply (blast dest!: Filter_Int_not_empty)
done
(*--------------------------------------------------------------------------------
This is a very long and tedious proof; need to break it into parts.
Have proof that {X. X <= S & (\<exists>f \<in> F. A Int f <= X)} is a filter as
a lemma
--------------------------------------------------------------------------------*)
lemma max_Filter_Ultrafilter:
"[| F \<in> Filter S;
\<forall>G \<in> Filter S. F <= G --> F = G |] ==> F \<in> Ultrafilter S"
apply (simp add: Ultrafilter_def)
apply (safe)
apply (rule ccontr)
apply (frule mem_FiltersetD [THEN is_FilterD2])
apply (frule_tac x = "{X. X <= S & (\<exists>f \<in> F. A Int f <= X) }" in bspec)
apply (rule mem_FiltersetI2)
apply (blast intro: elim:);
apply (simp add: );
apply (blast dest: mem_FiltersetD3)
apply (erule lemma_set_extend [THEN exE])
apply (assumption , erule lemma_set_not_empty)
txt{*First we prove @{term "{} \<notin> {X. X \<subseteq> S \<and> (\<exists>f\<in>F. A \<inter> f \<subseteq> X)}"}*}
apply (clarify );
apply (drule lemma_empty_Int_subset_Compl)
apply (frule mem_Filterset_disjI)
apply assumption;
apply (blast intro: elim:);
apply (fast dest: mem_FiltersetD3 elim:)
txt{*Next case: @{term "u \<inter> v"} is an element*}
apply (intro ballI)
apply (simp add: );
apply (rule conjI, blast)
apply (clarify );
apply (rule_tac x = "f Int fa" in bexI)
apply (fast intro: elim:);
apply (blast dest: mem_FiltersetD1 elim:)
apply force;
apply (blast dest: mem_FiltersetD3 elim:)
done
lemma Ultrafilter_iff: "(F \<in> Ultrafilter S) = (F \<in> Filter S & (\<forall>G \<in> Filter S. F <= G --> F = G))"
apply (blast intro!: Ultrafilter_max_Filter max_Filter_Ultrafilter)
done
subsection{* A Few Properties of Freefilters*}
lemma lemma_Compl_cancel_eq: "F1 Int F2 = ((F1 Int Y) Int F2) Un ((F2 Int (- Y)) Int F1)"
apply auto
done
lemma finite_IntI1: "finite X ==> finite (X Int Y)"
apply (erule Int_lower1 [THEN finite_subset])
done
lemma finite_IntI2: "finite Y ==> finite (X Int Y)"
apply (erule Int_lower2 [THEN finite_subset])
done
lemma finite_Int_Compl_cancel: "[| finite (F1 Int Y);
finite (F2 Int (- Y))
|] ==> finite (F1 Int F2)"
apply (rule_tac Y1 = "Y" in lemma_Compl_cancel_eq [THEN ssubst])
apply (rule finite_UnI)
apply (auto intro!: finite_IntI1 finite_IntI2)
done
lemma Freefilter_lemma_not_finite: "U \<in> Freefilter S ==>
~ (\<exists>f1 \<in> U. \<exists>f2 \<in> U. finite (f1 Int x)
& finite (f2 Int (- x)))"
apply (safe)
apply (frule_tac A = "f1" and B = "f2" in Freefilter_Filter [THEN mem_FiltersetD1])
apply (drule_tac [3] x = "f1 Int f2" in mem_FreefiltersetD1)
apply (drule_tac [4] finite_Int_Compl_cancel)
apply auto
done
(* the lemmas below follow *)
lemma Freefilter_Compl_not_finite_disjI: "U \<in> Freefilter S ==>
\<forall>f \<in> U. ~ finite (f Int x) | ~finite (f Int (- x))"
by (blast dest!: Freefilter_lemma_not_finite bspec)
lemma Freefilter_Compl_not_finite_disjI2: "U \<in> Freefilter S ==> (\<forall>f \<in> U. ~ finite (f Int x)) | (\<forall>f \<in> U. ~finite (f Int (- x)))"
apply (blast dest!: Freefilter_lemma_not_finite bspec)
done
lemma cofinite_Filter: "~ finite (UNIV:: 'a set) ==> {A:: 'a set. finite (- A)} \<in> Filter UNIV"
apply (rule mem_FiltersetI2)
apply (auto simp del: Collect_empty_eq)
apply (erule_tac c = "UNIV" in equalityCE)
apply auto
apply (erule Compl_anti_mono [THEN finite_subset])
apply assumption
done
lemma not_finite_UNIV_disjI: "~finite(UNIV :: 'a set) ==> ~finite (X :: 'a set) | ~finite (- X)"
apply (drule_tac A1 = "X" in Compl_partition [THEN ssubst])
apply simp
done
lemma not_finite_UNIV_Compl: "[| ~finite(UNIV :: 'a set); finite (X :: 'a set) |] ==> ~finite (- X)"
apply (drule_tac X = "X" in not_finite_UNIV_disjI)
apply blast
done
lemma mem_cofinite_Filter_not_finite:
"~ finite (UNIV:: 'a set)
==> \<forall>X \<in> {A:: 'a set. finite (- A)}. ~ finite X"
by (auto dest: not_finite_UNIV_disjI)
lemma cofinite_Freefilter:
"~ finite (UNIV:: 'a set) ==> {A:: 'a set. finite (- A)} \<in> Freefilter UNIV"
apply (rule mem_FreefiltersetI2)
apply (rule cofinite_Filter , assumption)
apply (blast dest!: mem_cofinite_Filter_not_finite)
done
(*????Set.thy*)
lemma UNIV_diff_Compl [simp]: "UNIV - x = - x"
apply auto
done
lemma FreeUltrafilter_contains_cofinite_set:
"[| ~finite(UNIV :: 'a set); (U :: 'a set set): FreeUltrafilter UNIV
|] ==> {X. finite(- X)} <= U"
by (auto simp add: Ultrafilter_def FreeUltrafilter_def)
(*--------------------------------------------------------------------
We prove: 1. Existence of maximal filter i.e. ultrafilter
2. Freeness property i.e ultrafilter is free
Use a locale to prove various lemmas and then
export main result: The Ultrafilter Theorem
-------------------------------------------------------------------*)
lemma (in UFT) chain_Un_subset_Pow:
"!!(c :: 'a set set set). c \<in> chain (superfrechet S) ==> Union c <= Pow S"
apply (simp add: chain_def superfrechet_def frechet_def)
apply (blast dest: mem_FiltersetD3 elim:)
done
lemma (in UFT) mem_chain_psubset_empty:
"!!(c :: 'a set set set). c: chain (superfrechet S)
==> !x: c. {} < x"
by (auto simp add: chain_def Filter_def is_Filter_def superfrechet_def frechet_def)
lemma (in UFT) chain_Un_not_empty: "!!(c :: 'a set set set).
[| c: chain (superfrechet S);
c ~= {} |]
==> Union(c) ~= {}"
apply (drule mem_chain_psubset_empty)
apply (safe)
apply (drule bspec , assumption)
apply (auto dest: Union_upper bspec simp add: psubset_def)
done
lemma (in UFT) Filter_empty_not_mem_Un:
"!!(c :: 'a set set set). c \<in> chain (superfrechet S) ==> {} ~: Union(c)"
by (auto simp add: is_Filter_def Filter_def chain_def superfrechet_def)
lemma (in UFT) Filter_Un_Int: "c \<in> chain (superfrechet S)
==> \<forall>u \<in> Union(c). \<forall>v \<in> Union(c). u Int v \<in> Union(c)"
apply (safe)
apply (frule_tac x = "X" and y = "Xa" in chainD)
apply (assumption)+
apply (drule chainD2)
apply (erule disjE)
apply (rule_tac [2] X = "X" in UnionI)
apply (rule_tac X = "Xa" in UnionI)
apply (auto intro: mem_FiltersetD1 simp add: superfrechet_def)
done
lemma (in UFT) Filter_Un_subset: "c \<in> chain (superfrechet S)
==> \<forall>u v. u \<in> Union(c) &
(u :: 'a set) <= v & v <= S --> v \<in> Union(c)"
apply (safe)
apply (drule chainD2)
apply (drule subsetD , assumption)
apply (rule UnionI , assumption)
apply (auto intro: mem_FiltersetD2 simp add: superfrechet_def)
done
lemma (in UFT) lemma_mem_chain_Filter:
"!!(c :: 'a set set set).
[| c \<in> chain (superfrechet S);
x \<in> c
|] ==> x \<in> Filter S"
by (auto simp add: chain_def superfrechet_def)
lemma (in UFT) lemma_mem_chain_frechet_subset:
"!!(c :: 'a set set set).
[| c \<in> chain (superfrechet S);
x \<in> c
|] ==> frechet S <= x"
by (auto simp add: chain_def superfrechet_def)
lemma (in UFT) Un_chain_mem_cofinite_Filter_set: "!!(c :: 'a set set set).
[| c ~= {};
c \<in> chain (superfrechet (UNIV :: 'a set))
|] ==> Union c \<in> superfrechet (UNIV)"
apply (simp (no_asm) add: superfrechet_def frechet_def)
apply (safe)
apply (rule mem_FiltersetI2)
apply (erule chain_Un_subset_Pow)
apply (rule UnionI , assumption)
apply (erule lemma_mem_chain_Filter [THEN mem_FiltersetD4] , assumption)
apply (erule chain_Un_not_empty)
apply (erule_tac [2] Filter_empty_not_mem_Un)
apply (erule_tac [2] Filter_Un_Int)
apply (erule_tac [2] Filter_Un_subset)
apply (subgoal_tac [2] "xa \<in> frechet (UNIV) ")
apply (blast intro: elim:);
apply (rule UnionI)
apply assumption;
apply (rule lemma_mem_chain_frechet_subset [THEN subsetD])
apply (auto simp add: frechet_def)
done
lemma (in UFT) max_cofinite_Filter_Ex: "\<exists>U \<in> superfrechet (UNIV).
\<forall>G \<in> superfrechet (UNIV). U <= G --> U = G"
apply (rule Zorn_Lemma2)
apply (insert not_finite_UNIV [THEN cofinite_Filter])
apply (safe)
apply (rule_tac Q = "c={}" in excluded_middle [THEN disjE])
apply (rule_tac x = "Union c" in bexI , blast)
apply (rule Un_chain_mem_cofinite_Filter_set);
apply (auto simp add: superfrechet_def frechet_def)
done
lemma (in UFT) max_cofinite_Freefilter_Ex: "\<exists>U \<in> superfrechet UNIV. (
\<forall>G \<in> superfrechet UNIV. U <= G --> U = G)
& (\<forall>x \<in> U. ~finite x)"
apply (insert not_finite_UNIV [THEN UFT.max_cofinite_Filter_Ex]);
apply (safe)
apply (rule_tac x = "U" in bexI)
apply (auto simp add: superfrechet_def frechet_def)
apply (drule_tac c = "- x" in subsetD)
apply (simp (no_asm_simp))
apply (frule_tac A = "x" and B = "- x" in mem_FiltersetD1)
apply (drule_tac [3] Filter_empty_not_mem)
apply (auto );
done
text{*There exists a free ultrafilter on any infinite set*}
theorem (in UFT) FreeUltrafilter_ex: "\<exists>U. U \<in> FreeUltrafilter (UNIV :: 'a set)"
apply (simp add: FreeUltrafilter_def)
apply (insert not_finite_UNIV [THEN UFT.max_cofinite_Freefilter_Ex])
apply (simp add: superfrechet_def Ultrafilter_iff frechet_def)
apply (safe)
apply (rule_tac x = "U" in exI)
apply (safe)
apply blast
done
theorems FreeUltrafilter_Ex = UFT.FreeUltrafilter_ex
end