(* Title : Filter.ML
ID : $Id$
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Description : Filters and Ultrafilter
*)
(*------------------------------------------------------------------
Properties of Filters and Freefilters -
rules for intro, destruction etc.
------------------------------------------------------------------*)
Goalw [is_Filter_def] "is_Filter X S ==> X <= Pow(S)";
by (Blast_tac 1);
qed "is_FilterD1";
Goalw [is_Filter_def] "is_Filter X S ==> X ~= {}";
by (Blast_tac 1);
qed "is_FilterD2";
Goalw [is_Filter_def] "is_Filter X S ==> {} ~: X";
by (Blast_tac 1);
qed "is_FilterD3";
Goalw [Filter_def] "is_Filter X S ==> X : Filter S";
by (Blast_tac 1);
qed "mem_FiltersetI";
Goalw [Filter_def] "X : Filter S ==> is_Filter X S";
by (Blast_tac 1);
qed "mem_FiltersetD";
Goal "X : Filter S ==> {} ~: X";
by (etac (mem_FiltersetD RS is_FilterD3) 1);
qed "Filter_empty_not_mem";
bind_thm ("Filter_empty_not_memE",(Filter_empty_not_mem RS notE));
Goalw [Filter_def,is_Filter_def]
"[| X: Filter S; A: X; B: X |] ==> A Int B : X";
by (Blast_tac 1);
qed "mem_FiltersetD1";
Goalw [Filter_def,is_Filter_def]
"[| X: Filter S; A: X; A <= B; B <= S|] ==> B : X";
by (Blast_tac 1);
qed "mem_FiltersetD2";
Goalw [Filter_def,is_Filter_def]
"[| X: Filter S; A: X |] ==> A : Pow S";
by (Blast_tac 1);
qed "mem_FiltersetD3";
Goalw [Filter_def,is_Filter_def]
"X: Filter S ==> S : X";
by (Blast_tac 1);
qed "mem_FiltersetD4";
Goalw [is_Filter_def]
"[| X <= Pow(S);\
\ S : X; \
\ X ~= {}; \
\ {} ~: X; \
\ ALL u: X. ALL v: X. u Int v : X; \
\ ALL u v. u: X & u<=v & v<=S --> v: X \
\ |] ==> is_Filter X S";
by (Blast_tac 1);
qed "is_FilterI";
Goal "[| X <= Pow(S);\
\ S : X; \
\ X ~= {}; \
\ {} ~: X; \
\ ALL u: X. ALL v: X. u Int v : X; \
\ ALL u v. u: X & u<=v & v<=S --> v: X \
\ |] ==> X: Filter S";
by (blast_tac (claset() addIs [mem_FiltersetI,is_FilterI]) 1);
qed "mem_FiltersetI2";
Goalw [is_Filter_def]
"is_Filter X S ==> X <= Pow(S) & \
\ S : X & \
\ X ~= {} & \
\ {} ~: X & \
\ (ALL u: X. ALL v: X. u Int v : X) & \
\ (ALL u v. u: X & u <= v & v<=S --> v: X)";
by (Fast_tac 1);
qed "is_FilterE_lemma";
Goalw [is_Filter_def]
"X : Filter S ==> X <= Pow(S) &\
\ S : X & \
\ X ~= {} & \
\ {} ~: X & \
\ (ALL u: X. ALL v: X. u Int v : X) & \
\ (ALL u v. u: X & u <= v & v<=S --> v: X)";
by (etac (mem_FiltersetD RS is_FilterE_lemma) 1);
qed "memFiltersetE_lemma";
Goalw [Filter_def,Freefilter_def]
"X: Freefilter S ==> X: Filter S";
by (Fast_tac 1);
qed "Freefilter_Filter";
Goalw [Freefilter_def]
"X: Freefilter S ==> ALL y: X. ~finite(y)";
by (Blast_tac 1);
qed "mem_Freefilter_not_finite";
Goal "[| X: Freefilter S; x: X |] ==> ~ finite x";
by (blast_tac (claset() addSDs [mem_Freefilter_not_finite]) 1);
qed "mem_FreefiltersetD1";
bind_thm ("mem_FreefiltersetE1", (mem_FreefiltersetD1 RS notE));
Goal "[| X: Freefilter S; finite x|] ==> x ~: X";
by (blast_tac (claset() addSDs [mem_Freefilter_not_finite]) 1);
qed "mem_FreefiltersetD2";
Goalw [Freefilter_def]
"[| X: Filter S; ALL x. ~(x: X & finite x) |] ==> X: Freefilter S";
by (Blast_tac 1);
qed "mem_FreefiltersetI1";
Goalw [Freefilter_def]
"[| X: Filter S; ALL x. (x ~: X | ~ finite x) |] ==> X: Freefilter S";
by (Blast_tac 1);
qed "mem_FreefiltersetI2";
Goal "[| X: Filter S; A: X; B: X |] ==> A Int B ~= {}";
by (forw_inst_tac [("A","A"),("B","B")] mem_FiltersetD1 1);
by (auto_tac (claset() addSDs [Filter_empty_not_mem],simpset()));
qed "Filter_Int_not_empty";
bind_thm ("Filter_Int_not_emptyE",(Filter_Int_not_empty RS notE));
(*----------------------------------------------------------------------------------
Ultrafilters and Free ultrafilters
----------------------------------------------------------------------------------*)
Goalw [Ultrafilter_def] "X : Ultrafilter S ==> X: Filter S";
by (Blast_tac 1);
qed "Ultrafilter_Filter";
Goalw [Ultrafilter_def]
"X : Ultrafilter S ==> !A: Pow(S). A : X | S - A: X";
by (Blast_tac 1);
qed "mem_UltrafiltersetD2";
Goalw [Ultrafilter_def]
"[|X : Ultrafilter S; A <= S; A ~: X |] ==> S - A: X";
by (Blast_tac 1);
qed "mem_UltrafiltersetD3";
Goalw [Ultrafilter_def]
"[|X : Ultrafilter S; A <= S; S - A ~: X |] ==> A: X";
by (Blast_tac 1);
qed "mem_UltrafiltersetD4";
Goalw [Ultrafilter_def]
"[| X: Filter S; \
\ ALL A: Pow(S). A: X | S - A : X |] ==> X: Ultrafilter S";
by (Blast_tac 1);
qed "mem_UltrafiltersetI";
Goalw [Ultrafilter_def,FreeUltrafilter_def]
"X: FreeUltrafilter S ==> X: Ultrafilter S";
by (Blast_tac 1);
qed "FreeUltrafilter_Ultrafilter";
Goalw [FreeUltrafilter_def]
"X: FreeUltrafilter S ==> ALL y: X. ~finite(y)";
by (Blast_tac 1);
qed "mem_FreeUltrafilter_not_finite";
Goal "[| X: FreeUltrafilter S; x: X |] ==> ~ finite x";
by (blast_tac (claset() addSDs [mem_FreeUltrafilter_not_finite]) 1);
qed "mem_FreeUltrafiltersetD1";
bind_thm ("mem_FreeUltrafiltersetE1", (mem_FreeUltrafiltersetD1 RS notE));
Goal "[| X: FreeUltrafilter S; finite x|] ==> x ~: X";
by (blast_tac (claset() addSDs [mem_FreeUltrafilter_not_finite]) 1);
qed "mem_FreeUltrafiltersetD2";
Goalw [FreeUltrafilter_def]
"[| X: Ultrafilter S; \
\ ALL x. ~(x: X & finite x) |] ==> X: FreeUltrafilter S";
by (Blast_tac 1);
qed "mem_FreeUltrafiltersetI1";
Goalw [FreeUltrafilter_def]
"[| X: Ultrafilter S; \
\ ALL x. (x ~: X | ~ finite x) |] ==> X: FreeUltrafilter S";
by (Blast_tac 1);
qed "mem_FreeUltrafiltersetI2";
Goalw [FreeUltrafilter_def,Freefilter_def,Ultrafilter_def]
"(X: FreeUltrafilter S) = (X: Freefilter S & (ALL x:Pow(S). x: X | S - x: X))";
by (Blast_tac 1);
qed "FreeUltrafilter_iff";
(*-------------------------------------------------------------------
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
---------------------------------------------------------------------*)
Goal "[| F ~= {}; A <= S |] ==> \
\ EX x. x: {X. X <= S & (EX f:F. A Int f <= X)}";
by (Blast_tac 1);
qed "lemma_set_extend";
Goal "a: X ==> X ~= {}";
by (Step_tac 1);
qed "lemma_set_not_empty";
Goal "x Int F <= {} ==> F <= - x";
by (Blast_tac 1);
qed "lemma_empty_Int_subset_Compl";
Goalw [Filter_def,is_Filter_def]
"[| F: Filter S; A ~: F; A <= S|] \
\ ==> ALL B. B ~: F | ~ B <= A";
by (Blast_tac 1);
qed "mem_Filterset_disjI";
Goal "F : Ultrafilter S ==> \
\ (F: Filter S & (ALL G: Filter S. F <= G --> F = G))";
by (auto_tac (claset(),simpset() addsimps [Ultrafilter_def]));
by (dres_inst_tac [("x","x")] bspec 1);
by (etac mem_FiltersetD3 1 THEN assume_tac 1);
by (Step_tac 1);
by (dtac subsetD 1 THEN assume_tac 1);
by (blast_tac (claset() addSDs [Filter_Int_not_empty]) 1);
qed "Ultrafilter_max_Filter";
(*--------------------------------------------------------------------------------
This is a very long and tedious proof; need to break it into parts.
Have proof that {X. X <= S & (EX f: F. A Int f <= X)} is a filter as
a lemma
--------------------------------------------------------------------------------*)
Goalw [Ultrafilter_def]
"[| F: Filter S; \
\ ALL G: Filter S. F <= G --> F = G |] ==> F : Ultrafilter S";
by (Step_tac 1);
by (rtac ccontr 1);
by (forward_tac [mem_FiltersetD RS is_FilterD2] 1);
by (forw_inst_tac [("x","{X. X <= S & (EX f: F. A Int f <= X)}")] bspec 1);
by (EVERY1[rtac mem_FiltersetI2, Blast_tac, Asm_full_simp_tac]);
by (blast_tac (claset() addDs [mem_FiltersetD3]) 1);
by (etac (lemma_set_extend RS exE) 1);
by (assume_tac 1 THEN etac lemma_set_not_empty 1);
by (REPEAT(rtac ballI 2) THEN Asm_full_simp_tac 2);
by (rtac conjI 2 THEN Blast_tac 2);
by (REPEAT(etac conjE 2) THEN REPEAT(etac bexE 2));
by (res_inst_tac [("x","f Int fa")] bexI 2);
by (etac mem_FiltersetD1 3);
by (assume_tac 3 THEN assume_tac 3);
by (Fast_tac 2);
by (EVERY[REPEAT(rtac allI 2), rtac impI 2,Asm_full_simp_tac 2]);
by (EVERY[REPEAT(etac conjE 2), etac bexE 2]);
by (res_inst_tac [("x","f")] bexI 2);
by (rtac subsetI 2);
by (Fast_tac 2 THEN assume_tac 2);
by (Step_tac 2);
by (Blast_tac 3);
by (eres_inst_tac [("c","A")] equalityCE 3);
by (REPEAT(Blast_tac 3));
by (dres_inst_tac [("A","xa")] mem_FiltersetD3 2 THEN assume_tac 2);
by (Blast_tac 2);
by (dtac lemma_empty_Int_subset_Compl 1);
by (EVERY1[ftac mem_Filterset_disjI , assume_tac, Fast_tac]);
by (dtac mem_FiltersetD3 1 THEN assume_tac 1);
by (dres_inst_tac [("x","f")] spec 1);
by (Blast_tac 1);
qed "max_Filter_Ultrafilter";
Goal "(F : Ultrafilter S) = (F: Filter S & (ALL G: Filter S. F <= G --> F = G))";
by (blast_tac (claset() addSIs [Ultrafilter_max_Filter,max_Filter_Ultrafilter]) 1);
qed "Ultrafilter_iff";
(*--------------------------------------------------------------------
A few properties of freefilters
-------------------------------------------------------------------*)
Goal "F1 Int F2 = ((F1 Int Y) Int F2) Un ((F2 Int (- Y)) Int F1)";
by (Auto_tac);
qed "lemma_Compl_cancel_eq";
Goal "finite X ==> finite (X Int Y)";
by (etac (Int_lower1 RS finite_subset) 1);
qed "finite_IntI1";
Goal "finite Y ==> finite (X Int Y)";
by (etac (Int_lower2 RS finite_subset) 1);
qed "finite_IntI2";
Goal "[| finite (F1 Int Y); \
\ finite (F2 Int (- Y)) \
\ |] ==> finite (F1 Int F2)";
by (res_inst_tac [("Y1","Y")] (lemma_Compl_cancel_eq RS ssubst) 1);
by (rtac finite_UnI 1);
by (auto_tac (claset() addSIs [finite_IntI1,finite_IntI2],simpset()));
qed "finite_Int_Compl_cancel";
Goal "U: Freefilter S ==> \
\ ~ (EX f1: U. EX f2: U. finite (f1 Int x) \
\ & finite (f2 Int (- x)))";
by (Step_tac 1);
by (forw_inst_tac [("A","f1"),("B","f2")]
(Freefilter_Filter RS mem_FiltersetD1) 1);
by (dres_inst_tac [("x","f1 Int f2")] mem_FreefiltersetD1 3);
by (dtac finite_Int_Compl_cancel 4);
by (Auto_tac);
qed "Freefilter_lemma_not_finite";
(* the lemmas below follow *)
Goal "U: Freefilter S ==> \
\ ALL f: U. ~ finite (f Int x) | ~finite (f Int (- x))";
by (blast_tac (claset() addSDs [Freefilter_lemma_not_finite,bspec]) 1);
qed "Freefilter_Compl_not_finite_disjI";
Goal "U: Freefilter S ==> \
\ (ALL f: U. ~ finite (f Int x)) | (ALL f:U. ~finite (f Int (- x)))";
by (blast_tac (claset() addSDs [Freefilter_lemma_not_finite,bspec]) 1);
qed "Freefilter_Compl_not_finite_disjI2";
Goal "- UNIV = {}";
by (Auto_tac );
qed "Compl_UNIV_eq";
Addsimps [Compl_UNIV_eq];
Goal "- {} = UNIV";
by (Auto_tac );
qed "Compl_empty_eq";
Addsimps [Compl_empty_eq];
val [prem] = goal thy "~ finite (UNIV:: 'a set) ==> \
\ {A:: 'a set. finite (- A)} : Filter UNIV";
by (cut_facts_tac [prem] 1);
by (rtac mem_FiltersetI2 1);
by (auto_tac (claset(), simpset() delsimps [Collect_empty_eq]));
by (eres_inst_tac [("c","UNIV")] equalityCE 1);
by (Auto_tac);
by (etac (Compl_anti_mono RS finite_subset) 1);
by (assume_tac 1);
qed "cofinite_Filter";
Goal "~finite(UNIV :: 'a set) ==> ~finite (X :: 'a set) | ~finite (- X)";
by (dres_inst_tac [("A1","X")] (Compl_partition RS ssubst) 1);
by (Asm_full_simp_tac 1);
qed "not_finite_UNIV_disjI";
Goal "[| ~finite(UNIV :: 'a set); \
\ finite (X :: 'a set) \
\ |] ==> ~finite (- X)";
by (dres_inst_tac [("X","X")] not_finite_UNIV_disjI 1);
by (Blast_tac 1);
qed "not_finite_UNIV_Compl";
val [prem] = goal thy "~ finite (UNIV:: 'a set) ==> \
\ !X: {A:: 'a set. finite (- A)}. ~ finite X";
by (cut_facts_tac [prem] 1);
by (auto_tac (claset() addDs [not_finite_UNIV_disjI],simpset()));
qed "mem_cofinite_Filter_not_finite";
val [prem] = goal thy "~ finite (UNIV:: 'a set) ==> \
\ {A:: 'a set. finite (- A)} : Freefilter UNIV";
by (cut_facts_tac [prem] 1);
by (rtac mem_FreefiltersetI2 1);
by (rtac cofinite_Filter 1 THEN assume_tac 1);
by (blast_tac (claset() addSDs [mem_cofinite_Filter_not_finite]) 1);
qed "cofinite_Freefilter";
Goal "UNIV - x = - x";
by (Auto_tac);
qed "UNIV_diff_Compl";
Addsimps [UNIV_diff_Compl];
Goalw [Ultrafilter_def,FreeUltrafilter_def]
"[| ~finite(UNIV :: 'a set); (U :: 'a set set): FreeUltrafilter UNIV\
\ |] ==> {X. finite(- X)} <= U";
by (ftac cofinite_Filter 1);
by (Step_tac 1);
by (forw_inst_tac [("X","- x :: 'a set")] not_finite_UNIV_Compl 1);
by (assume_tac 1);
by (Step_tac 1 THEN Fast_tac 1);
by (dres_inst_tac [("x","x")] bspec 1);
by (Blast_tac 1);
by (asm_full_simp_tac (simpset() addsimps [UNIV_diff_Compl]) 1);
qed "FreeUltrafilter_contains_cofinite_set";
(*--------------------------------------------------------------------
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
-------------------------------------------------------------------*)
Open_locale "UFT";
Goalw [chain_def, thm "superfrechet_def", thm "frechet_def"]
"!!(c :: 'a set set set). c : chain (superfrechet S) ==> Union c <= Pow S";
by (Step_tac 1);
by (dtac subsetD 1 THEN assume_tac 1);
by (Step_tac 1);
by (dres_inst_tac [("X","X")] mem_FiltersetD3 1);
by (Auto_tac);
qed "chain_Un_subset_Pow";
Goalw [chain_def,Filter_def,is_Filter_def,
thm "superfrechet_def", thm "frechet_def"]
"!!(c :: 'a set set set). c: chain (superfrechet S) \
\ ==> !x: c. {} < x";
by (blast_tac (claset() addSIs [psubsetI]) 1);
qed "mem_chain_psubset_empty";
Goal "!!(c :: 'a set set set). \
\ [| c: chain (superfrechet S);\
\ c ~= {} \
\ |]\
\ ==> Union(c) ~= {}";
by (dtac mem_chain_psubset_empty 1);
by (Step_tac 1);
by (dtac bspec 1 THEN assume_tac 1);
by (auto_tac (claset() addDs [Union_upper,bspec],
simpset() addsimps [psubset_def]));
qed "chain_Un_not_empty";
Goalw [is_Filter_def,Filter_def,chain_def,thm "superfrechet_def"]
"!!(c :: 'a set set set). \
\ c : chain (superfrechet S) \
\ ==> {} ~: Union(c)";
by (Blast_tac 1);
qed "Filter_empty_not_mem_Un";
Goal "c: chain (superfrechet S) \
\ ==> ALL u : Union(c). ALL v: Union(c). u Int v : Union(c)";
by (Step_tac 1);
by (forw_inst_tac [("x","X"),("y","Xa")] chainD 1);
by (REPEAT(assume_tac 1));
by (dtac chainD2 1);
by (etac disjE 1);
by (res_inst_tac [("X","Xa")] UnionI 1 THEN assume_tac 1);
by (dres_inst_tac [("A","X")] subsetD 1 THEN assume_tac 1);
by (dres_inst_tac [("c","Xa")] subsetD 1 THEN assume_tac 1);
by (res_inst_tac [("X","X")] UnionI 2 THEN assume_tac 2);
by (dres_inst_tac [("A","Xa")] subsetD 2 THEN assume_tac 2);
by (dres_inst_tac [("c","X")] subsetD 2 THEN assume_tac 2);
by (auto_tac (claset() addIs [mem_FiltersetD1],
simpset() addsimps [thm "superfrechet_def"]));
qed "Filter_Un_Int";
Goal "c: chain (superfrechet S) \
\ ==> ALL u v. u: Union(c) & \
\ (u :: 'a set) <= v & v <= S --> v: Union(c)";
by (Step_tac 1);
by (dtac chainD2 1);
by (dtac subsetD 1 THEN assume_tac 1);
by (rtac UnionI 1 THEN assume_tac 1);
by (auto_tac (claset() addIs [mem_FiltersetD2],
simpset() addsimps [thm "superfrechet_def"]));
qed "Filter_Un_subset";
Goalw [chain_def,thm "superfrechet_def"]
"!!(c :: 'a set set set). \
\ [| c: chain (superfrechet S);\
\ x: c \
\ |] ==> x : Filter S";
by (Blast_tac 1);
qed "lemma_mem_chain_Filter";
Goalw [chain_def,thm "superfrechet_def"]
"!!(c :: 'a set set set). \
\ [| c: chain (superfrechet S);\
\ x: c \
\ |] ==> frechet S <= x";
by (Blast_tac 1);
qed "lemma_mem_chain_frechet_subset";
Goal "!!(c :: 'a set set set). \
\ [| c ~= {}; \
\ c : chain (superfrechet (UNIV :: 'a set))\
\ |] ==> Union c : superfrechet (UNIV)";
by (simp_tac (simpset() addsimps
[thm "superfrechet_def",thm "frechet_def"]) 1);
by (Step_tac 1);
by (rtac mem_FiltersetI2 1);
by (etac chain_Un_subset_Pow 1);
by (rtac UnionI 1 THEN assume_tac 1);
by (etac (lemma_mem_chain_Filter RS mem_FiltersetD4) 1 THEN assume_tac 1);
by (etac chain_Un_not_empty 1);
by (etac Filter_empty_not_mem_Un 2);
by (etac Filter_Un_Int 2);
by (etac Filter_Un_subset 2);
by (subgoal_tac "xa : frechet (UNIV)" 2);
by (rtac UnionI 2 THEN assume_tac 2);
by (rtac (lemma_mem_chain_frechet_subset RS subsetD) 2);
by (auto_tac (claset(),simpset() addsimps [thm "frechet_def"]));
qed "Un_chain_mem_cofinite_Filter_set";
Goal "EX U: superfrechet (UNIV). \
\ ALL G: superfrechet (UNIV). U <= G --> U = G";
by (rtac Zorn_Lemma2 1);
by (cut_facts_tac [thm "not_finite_UNIV" RS cofinite_Filter] 1);
by (Step_tac 1);
by (res_inst_tac [("Q","c={}")] (excluded_middle RS disjE) 1);
by (res_inst_tac [("x","Union c")] bexI 1 THEN Blast_tac 1);
by (rtac Un_chain_mem_cofinite_Filter_set 1 THEN REPEAT(assume_tac 1));
by (res_inst_tac [("x","frechet (UNIV)")] bexI 1 THEN Blast_tac 1);
by (auto_tac (claset(),
simpset() addsimps
[thm "superfrechet_def", thm "frechet_def"]));
qed "max_cofinite_Filter_Ex";
Goal "EX U: superfrechet UNIV. (\
\ ALL G: superfrechet UNIV. U <= G --> U = G) \
\ & (ALL x: U. ~finite x)";
by (cut_facts_tac [thm "not_finite_UNIV" RS
(export max_cofinite_Filter_Ex)] 1);
by (Step_tac 1);
by (res_inst_tac [("x","U")] bexI 1);
by (auto_tac (claset(),simpset() addsimps
[thm "superfrechet_def", thm "frechet_def"]));
by (dres_inst_tac [("c","- x")] subsetD 1);
by (Asm_simp_tac 1);
by (forw_inst_tac [("A","x"),("B","- x")] mem_FiltersetD1 1);
by (dtac Filter_empty_not_mem 3);
by (ALLGOALS(Asm_full_simp_tac ));
qed "max_cofinite_Freefilter_Ex";
(*--------------------------------------------------------------------------------
There exists a free ultrafilter on any infinite set
--------------------------------------------------------------------------------*)
Goalw [FreeUltrafilter_def] "EX U. U: FreeUltrafilter (UNIV :: 'a set)";
by (cut_facts_tac [thm "not_finite_UNIV" RS (export max_cofinite_Freefilter_Ex)] 1);
by (asm_full_simp_tac (simpset() addsimps
[thm "superfrechet_def", Ultrafilter_iff, thm "frechet_def"]) 1);
by (Step_tac 1);
by (res_inst_tac [("x","U")] exI 1);
by (Step_tac 1);
by (Blast_tac 1);
qed "FreeUltrafilter_ex";
val FreeUltrafilter_Ex = export FreeUltrafilter_ex;
Close_locale "UFT";