author | paulson |
Sat, 31 Jul 2004 20:54:23 +0200 | |
changeset 15094 | a7d1a3fdc30d |
parent 13551 | b7f64ee8da84 |
permissions | -rw-r--r-- |
10750 | 1 |
(* Title : Filter.ML |
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ID : $Id$ |
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Author : Jacques D. Fleuriot |
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Copyright : 1998 University of Cambridge |
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Description : Filters and Ultrafilter |
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*) |
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converted Hyperreal/Zorn to Isar format and moved to Library
paulson
parents:
10750
diff
changeset
|
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(*ML bindings for Library/Zorn theorems*) |
b7f64ee8da84
converted Hyperreal/Zorn to Isar format and moved to Library
paulson
parents:
10750
diff
changeset
|
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val chain_def = thm "chain_def"; |
b7f64ee8da84
converted Hyperreal/Zorn to Isar format and moved to Library
paulson
parents:
10750
diff
changeset
|
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val chainD = thm "chainD"; |
b7f64ee8da84
converted Hyperreal/Zorn to Isar format and moved to Library
paulson
parents:
10750
diff
changeset
|
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val chainD2 = thm "chainD2"; |
b7f64ee8da84
converted Hyperreal/Zorn to Isar format and moved to Library
paulson
parents:
10750
diff
changeset
|
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val Zorn_Lemma2 = thm "Zorn_Lemma2"; |
b7f64ee8da84
converted Hyperreal/Zorn to Isar format and moved to Library
paulson
parents:
10750
diff
changeset
|
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10750 | 14 |
(*------------------------------------------------------------------ |
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Properties of Filters and Freefilters - |
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rules for intro, destruction etc. |
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------------------------------------------------------------------*) |
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Goalw [is_Filter_def] "is_Filter X S ==> X <= Pow(S)"; |
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by (Blast_tac 1); |
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qed "is_FilterD1"; |
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||
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Goalw [is_Filter_def] "is_Filter X S ==> X ~= {}"; |
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by (Blast_tac 1); |
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qed "is_FilterD2"; |
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||
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Goalw [is_Filter_def] "is_Filter X S ==> {} ~: X"; |
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by (Blast_tac 1); |
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qed "is_FilterD3"; |
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||
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Goalw [Filter_def] "is_Filter X S ==> X : Filter S"; |
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by (Blast_tac 1); |
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qed "mem_FiltersetI"; |
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||
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Goalw [Filter_def] "X : Filter S ==> is_Filter X S"; |
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by (Blast_tac 1); |
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qed "mem_FiltersetD"; |
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||
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Goal "X : Filter S ==> {} ~: X"; |
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by (etac (mem_FiltersetD RS is_FilterD3) 1); |
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qed "Filter_empty_not_mem"; |
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bind_thm ("Filter_empty_not_memE",(Filter_empty_not_mem RS notE)); |
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||
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Goalw [Filter_def,is_Filter_def] |
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"[| X: Filter S; A: X; B: X |] ==> A Int B : X"; |
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by (Blast_tac 1); |
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qed "mem_FiltersetD1"; |
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||
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Goalw [Filter_def,is_Filter_def] |
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"[| X: Filter S; A: X; A <= B; B <= S|] ==> B : X"; |
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by (Blast_tac 1); |
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qed "mem_FiltersetD2"; |
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||
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Goalw [Filter_def,is_Filter_def] |
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"[| X: Filter S; A: X |] ==> A : Pow S"; |
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by (Blast_tac 1); |
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qed "mem_FiltersetD3"; |
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||
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Goalw [Filter_def,is_Filter_def] |
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"X: Filter S ==> S : X"; |
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by (Blast_tac 1); |
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qed "mem_FiltersetD4"; |
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||
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Goalw [is_Filter_def] |
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"[| X <= Pow(S);\ |
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\ S : X; \ |
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\ X ~= {}; \ |
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\ {} ~: X; \ |
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\ ALL u: X. ALL v: X. u Int v : X; \ |
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\ ALL u v. u: X & u<=v & v<=S --> v: X \ |
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\ |] ==> is_Filter X S"; |
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by (Blast_tac 1); |
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qed "is_FilterI"; |
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Goal "[| X <= Pow(S);\ |
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\ S : X; \ |
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\ X ~= {}; \ |
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\ {} ~: X; \ |
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\ ALL u: X. ALL v: X. u Int v : X; \ |
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\ ALL u v. u: X & u<=v & v<=S --> v: X \ |
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\ |] ==> X: Filter S"; |
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by (blast_tac (claset() addIs [mem_FiltersetI,is_FilterI]) 1); |
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qed "mem_FiltersetI2"; |
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||
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Goalw [is_Filter_def] |
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"is_Filter X S ==> X <= Pow(S) & \ |
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\ S : X & \ |
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\ X ~= {} & \ |
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\ {} ~: X & \ |
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\ (ALL u: X. ALL v: X. u Int v : X) & \ |
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\ (ALL u v. u: X & u <= v & v<=S --> v: X)"; |
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by (Fast_tac 1); |
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qed "is_FilterE_lemma"; |
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Goalw [is_Filter_def] |
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"X : Filter S ==> X <= Pow(S) &\ |
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\ S : X & \ |
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\ X ~= {} & \ |
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\ {} ~: X & \ |
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\ (ALL u: X. ALL v: X. u Int v : X) & \ |
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\ (ALL u v. u: X & u <= v & v<=S --> v: X)"; |
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by (etac (mem_FiltersetD RS is_FilterE_lemma) 1); |
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qed "memFiltersetE_lemma"; |
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||
106 |
Goalw [Filter_def,Freefilter_def] |
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"X: Freefilter S ==> X: Filter S"; |
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by (Fast_tac 1); |
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qed "Freefilter_Filter"; |
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111 |
Goalw [Freefilter_def] |
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"X: Freefilter S ==> ALL y: X. ~finite(y)"; |
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by (Blast_tac 1); |
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qed "mem_Freefilter_not_finite"; |
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Goal "[| X: Freefilter S; x: X |] ==> ~ finite x"; |
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by (blast_tac (claset() addSDs [mem_Freefilter_not_finite]) 1); |
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qed "mem_FreefiltersetD1"; |
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bind_thm ("mem_FreefiltersetE1", (mem_FreefiltersetD1 RS notE)); |
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Goal "[| X: Freefilter S; finite x|] ==> x ~: X"; |
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by (blast_tac (claset() addSDs [mem_Freefilter_not_finite]) 1); |
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qed "mem_FreefiltersetD2"; |
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Goalw [Freefilter_def] |
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"[| X: Filter S; ALL x. ~(x: X & finite x) |] ==> X: Freefilter S"; |
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by (Blast_tac 1); |
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qed "mem_FreefiltersetI1"; |
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||
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Goalw [Freefilter_def] |
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"[| X: Filter S; ALL x. (x ~: X | ~ finite x) |] ==> X: Freefilter S"; |
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by (Blast_tac 1); |
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qed "mem_FreefiltersetI2"; |
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Goal "[| X: Filter S; A: X; B: X |] ==> A Int B ~= {}"; |
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by (forw_inst_tac [("A","A"),("B","B")] mem_FiltersetD1 1); |
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by (auto_tac (claset() addSDs [Filter_empty_not_mem],simpset())); |
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qed "Filter_Int_not_empty"; |
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bind_thm ("Filter_Int_not_emptyE",(Filter_Int_not_empty RS notE)); |
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||
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(*---------------------------------------------------------------------------------- |
|
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Ultrafilters and Free ultrafilters |
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----------------------------------------------------------------------------------*) |
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||
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Goalw [Ultrafilter_def] "X : Ultrafilter S ==> X: Filter S"; |
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by (Blast_tac 1); |
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qed "Ultrafilter_Filter"; |
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||
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Goalw [Ultrafilter_def] |
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"X : Ultrafilter S ==> !A: Pow(S). A : X | S - A: X"; |
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by (Blast_tac 1); |
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qed "mem_UltrafiltersetD2"; |
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Goalw [Ultrafilter_def] |
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"[|X : Ultrafilter S; A <= S; A ~: X |] ==> S - A: X"; |
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by (Blast_tac 1); |
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qed "mem_UltrafiltersetD3"; |
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Goalw [Ultrafilter_def] |
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"[|X : Ultrafilter S; A <= S; S - A ~: X |] ==> A: X"; |
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by (Blast_tac 1); |
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qed "mem_UltrafiltersetD4"; |
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Goalw [Ultrafilter_def] |
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"[| X: Filter S; \ |
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\ ALL A: Pow(S). A: X | S - A : X |] ==> X: Ultrafilter S"; |
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by (Blast_tac 1); |
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qed "mem_UltrafiltersetI"; |
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Goalw [Ultrafilter_def,FreeUltrafilter_def] |
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"X: FreeUltrafilter S ==> X: Ultrafilter S"; |
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by (Blast_tac 1); |
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qed "FreeUltrafilter_Ultrafilter"; |
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Goalw [FreeUltrafilter_def] |
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"X: FreeUltrafilter S ==> ALL y: X. ~finite(y)"; |
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by (Blast_tac 1); |
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qed "mem_FreeUltrafilter_not_finite"; |
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Goal "[| X: FreeUltrafilter S; x: X |] ==> ~ finite x"; |
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by (blast_tac (claset() addSDs [mem_FreeUltrafilter_not_finite]) 1); |
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qed "mem_FreeUltrafiltersetD1"; |
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bind_thm ("mem_FreeUltrafiltersetE1", (mem_FreeUltrafiltersetD1 RS notE)); |
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Goal "[| X: FreeUltrafilter S; finite x|] ==> x ~: X"; |
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by (blast_tac (claset() addSDs [mem_FreeUltrafilter_not_finite]) 1); |
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qed "mem_FreeUltrafiltersetD2"; |
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Goalw [FreeUltrafilter_def] |
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"[| X: Ultrafilter S; \ |
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\ ALL x. ~(x: X & finite x) |] ==> X: FreeUltrafilter S"; |
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by (Blast_tac 1); |
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qed "mem_FreeUltrafiltersetI1"; |
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Goalw [FreeUltrafilter_def] |
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"[| X: Ultrafilter S; \ |
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\ ALL x. (x ~: X | ~ finite x) |] ==> X: FreeUltrafilter S"; |
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by (Blast_tac 1); |
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qed "mem_FreeUltrafiltersetI2"; |
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Goalw [FreeUltrafilter_def,Freefilter_def,Ultrafilter_def] |
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"(X: FreeUltrafilter S) = (X: Freefilter S & (ALL x:Pow(S). x: X | S - x: X))"; |
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by (Blast_tac 1); |
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qed "FreeUltrafilter_iff"; |
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||
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(*------------------------------------------------------------------- |
|
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A Filter F on S is an ultrafilter iff it is a maximal filter |
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i.e. whenever G is a filter on I and F <= F then F = G |
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--------------------------------------------------------------------*) |
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(*--------------------------------------------------------------------- |
|
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lemmas that shows existence of an extension to what was assumed to |
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be a maximal filter. Will be used to derive contradiction in proof of |
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property of ultrafilter |
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---------------------------------------------------------------------*) |
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Goal "[| F ~= {}; A <= S |] ==> \ |
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\ EX x. x: {X. X <= S & (EX f:F. A Int f <= X)}"; |
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by (Blast_tac 1); |
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qed "lemma_set_extend"; |
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223 |
Goal "a: X ==> X ~= {}"; |
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by (Step_tac 1); |
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qed "lemma_set_not_empty"; |
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226 |
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227 |
Goal "x Int F <= {} ==> F <= - x"; |
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by (Blast_tac 1); |
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qed "lemma_empty_Int_subset_Compl"; |
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Goalw [Filter_def,is_Filter_def] |
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"[| F: Filter S; A ~: F; A <= S|] \ |
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\ ==> ALL B. B ~: F | ~ B <= A"; |
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by (Blast_tac 1); |
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qed "mem_Filterset_disjI"; |
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Goal "F : Ultrafilter S ==> \ |
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\ (F: Filter S & (ALL G: Filter S. F <= G --> F = G))"; |
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by (auto_tac (claset(),simpset() addsimps [Ultrafilter_def])); |
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by (dres_inst_tac [("x","x")] bspec 1); |
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by (etac mem_FiltersetD3 1 THEN assume_tac 1); |
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by (Step_tac 1); |
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by (dtac subsetD 1 THEN assume_tac 1); |
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by (blast_tac (claset() addSDs [Filter_Int_not_empty]) 1); |
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qed "Ultrafilter_max_Filter"; |
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(*-------------------------------------------------------------------------------- |
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This is a very long and tedious proof; need to break it into parts. |
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Have proof that {X. X <= S & (EX f: F. A Int f <= X)} is a filter as |
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a lemma |
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--------------------------------------------------------------------------------*) |
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Goalw [Ultrafilter_def] |
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"[| F: Filter S; \ |
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\ ALL G: Filter S. F <= G --> F = G |] ==> F : Ultrafilter S"; |
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by (Step_tac 1); |
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by (rtac ccontr 1); |
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by (forward_tac [mem_FiltersetD RS is_FilterD2] 1); |
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by (forw_inst_tac [("x","{X. X <= S & (EX f: F. A Int f <= X)}")] bspec 1); |
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by (EVERY1[rtac mem_FiltersetI2, Blast_tac, Asm_full_simp_tac]); |
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by (blast_tac (claset() addDs [mem_FiltersetD3]) 1); |
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by (etac (lemma_set_extend RS exE) 1); |
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by (assume_tac 1 THEN etac lemma_set_not_empty 1); |
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by (REPEAT(rtac ballI 2) THEN Asm_full_simp_tac 2); |
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by (rtac conjI 2 THEN Blast_tac 2); |
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by (REPEAT(etac conjE 2) THEN REPEAT(etac bexE 2)); |
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by (res_inst_tac [("x","f Int fa")] bexI 2); |
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by (etac mem_FiltersetD1 3); |
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by (assume_tac 3 THEN assume_tac 3); |
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by (Fast_tac 2); |
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by (EVERY[REPEAT(rtac allI 2), rtac impI 2,Asm_full_simp_tac 2]); |
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by (EVERY[REPEAT(etac conjE 2), etac bexE 2]); |
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by (res_inst_tac [("x","f")] bexI 2); |
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by (rtac subsetI 2); |
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by (Fast_tac 2 THEN assume_tac 2); |
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by (Step_tac 2); |
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by (Blast_tac 3); |
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by (eres_inst_tac [("c","A")] equalityCE 3); |
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by (REPEAT(Blast_tac 3)); |
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by (dres_inst_tac [("A","xa")] mem_FiltersetD3 2 THEN assume_tac 2); |
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by (Blast_tac 2); |
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by (dtac lemma_empty_Int_subset_Compl 1); |
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by (EVERY1[ftac mem_Filterset_disjI , assume_tac, Fast_tac]); |
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by (dtac mem_FiltersetD3 1 THEN assume_tac 1); |
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by (dres_inst_tac [("x","f")] spec 1); |
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by (Blast_tac 1); |
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qed "max_Filter_Ultrafilter"; |
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288 |
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Goal "(F : Ultrafilter S) = (F: Filter S & (ALL G: Filter S. F <= G --> F = G))"; |
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by (blast_tac (claset() addSIs [Ultrafilter_max_Filter,max_Filter_Ultrafilter]) 1); |
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qed "Ultrafilter_iff"; |
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292 |
||
293 |
(*-------------------------------------------------------------------- |
|
294 |
A few properties of freefilters |
|
295 |
-------------------------------------------------------------------*) |
|
296 |
||
297 |
Goal "F1 Int F2 = ((F1 Int Y) Int F2) Un ((F2 Int (- Y)) Int F1)"; |
|
298 |
by (Auto_tac); |
|
299 |
qed "lemma_Compl_cancel_eq"; |
|
300 |
||
301 |
Goal "finite X ==> finite (X Int Y)"; |
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302 |
by (etac (Int_lower1 RS finite_subset) 1); |
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303 |
qed "finite_IntI1"; |
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304 |
||
305 |
Goal "finite Y ==> finite (X Int Y)"; |
|
306 |
by (etac (Int_lower2 RS finite_subset) 1); |
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307 |
qed "finite_IntI2"; |
|
308 |
||
309 |
Goal "[| finite (F1 Int Y); \ |
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310 |
\ finite (F2 Int (- Y)) \ |
|
311 |
\ |] ==> finite (F1 Int F2)"; |
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312 |
by (res_inst_tac [("Y1","Y")] (lemma_Compl_cancel_eq RS ssubst) 1); |
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313 |
by (rtac finite_UnI 1); |
|
314 |
by (auto_tac (claset() addSIs [finite_IntI1,finite_IntI2],simpset())); |
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315 |
qed "finite_Int_Compl_cancel"; |
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316 |
||
317 |
Goal "U: Freefilter S ==> \ |
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318 |
\ ~ (EX f1: U. EX f2: U. finite (f1 Int x) \ |
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319 |
\ & finite (f2 Int (- x)))"; |
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320 |
by (Step_tac 1); |
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321 |
by (forw_inst_tac [("A","f1"),("B","f2")] |
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322 |
(Freefilter_Filter RS mem_FiltersetD1) 1); |
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323 |
by (dres_inst_tac [("x","f1 Int f2")] mem_FreefiltersetD1 3); |
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324 |
by (dtac finite_Int_Compl_cancel 4); |
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325 |
by (Auto_tac); |
|
326 |
qed "Freefilter_lemma_not_finite"; |
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327 |
||
328 |
(* the lemmas below follow *) |
|
329 |
Goal "U: Freefilter S ==> \ |
|
330 |
\ ALL f: U. ~ finite (f Int x) | ~finite (f Int (- x))"; |
|
331 |
by (blast_tac (claset() addSDs [Freefilter_lemma_not_finite,bspec]) 1); |
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332 |
qed "Freefilter_Compl_not_finite_disjI"; |
|
333 |
||
334 |
Goal "U: Freefilter S ==> \ |
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335 |
\ (ALL f: U. ~ finite (f Int x)) | (ALL f:U. ~finite (f Int (- x)))"; |
|
336 |
by (blast_tac (claset() addSDs [Freefilter_lemma_not_finite,bspec]) 1); |
|
337 |
qed "Freefilter_Compl_not_finite_disjI2"; |
|
338 |
||
339 |
Goal "- UNIV = {}"; |
|
340 |
by (Auto_tac ); |
|
341 |
qed "Compl_UNIV_eq"; |
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342 |
||
343 |
Addsimps [Compl_UNIV_eq]; |
|
344 |
||
345 |
Goal "- {} = UNIV"; |
|
346 |
by (Auto_tac ); |
|
347 |
qed "Compl_empty_eq"; |
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348 |
||
349 |
Addsimps [Compl_empty_eq]; |
|
350 |
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351 |
val [prem] = goal (the_context ()) "~ finite (UNIV:: 'a set) ==> \ |
|
352 |
\ {A:: 'a set. finite (- A)} : Filter UNIV"; |
|
353 |
by (cut_facts_tac [prem] 1); |
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354 |
by (rtac mem_FiltersetI2 1); |
|
355 |
by (auto_tac (claset(), simpset() delsimps [Collect_empty_eq])); |
|
356 |
by (eres_inst_tac [("c","UNIV")] equalityCE 1); |
|
357 |
by (Auto_tac); |
|
358 |
by (etac (Compl_anti_mono RS finite_subset) 1); |
|
359 |
by (assume_tac 1); |
|
360 |
qed "cofinite_Filter"; |
|
361 |
||
362 |
Goal "~finite(UNIV :: 'a set) ==> ~finite (X :: 'a set) | ~finite (- X)"; |
|
363 |
by (dres_inst_tac [("A1","X")] (Compl_partition RS ssubst) 1); |
|
364 |
by (Asm_full_simp_tac 1); |
|
365 |
qed "not_finite_UNIV_disjI"; |
|
366 |
||
367 |
Goal "[| ~finite(UNIV :: 'a set); \ |
|
368 |
\ finite (X :: 'a set) \ |
|
369 |
\ |] ==> ~finite (- X)"; |
|
370 |
by (dres_inst_tac [("X","X")] not_finite_UNIV_disjI 1); |
|
371 |
by (Blast_tac 1); |
|
372 |
qed "not_finite_UNIV_Compl"; |
|
373 |
||
374 |
val [prem] = goal (the_context ()) "~ finite (UNIV:: 'a set) ==> \ |
|
375 |
\ !X: {A:: 'a set. finite (- A)}. ~ finite X"; |
|
376 |
by (cut_facts_tac [prem] 1); |
|
377 |
by (auto_tac (claset() addDs [not_finite_UNIV_disjI],simpset())); |
|
378 |
qed "mem_cofinite_Filter_not_finite"; |
|
379 |
||
380 |
val [prem] = goal (the_context ()) "~ finite (UNIV:: 'a set) ==> \ |
|
381 |
\ {A:: 'a set. finite (- A)} : Freefilter UNIV"; |
|
382 |
by (cut_facts_tac [prem] 1); |
|
383 |
by (rtac mem_FreefiltersetI2 1); |
|
384 |
by (rtac cofinite_Filter 1 THEN assume_tac 1); |
|
385 |
by (blast_tac (claset() addSDs [mem_cofinite_Filter_not_finite]) 1); |
|
386 |
qed "cofinite_Freefilter"; |
|
387 |
||
388 |
Goal "UNIV - x = - x"; |
|
389 |
by (Auto_tac); |
|
390 |
qed "UNIV_diff_Compl"; |
|
391 |
Addsimps [UNIV_diff_Compl]; |
|
392 |
||
393 |
Goalw [Ultrafilter_def,FreeUltrafilter_def] |
|
394 |
"[| ~finite(UNIV :: 'a set); (U :: 'a set set): FreeUltrafilter UNIV\ |
|
395 |
\ |] ==> {X. finite(- X)} <= U"; |
|
396 |
by (ftac cofinite_Filter 1); |
|
397 |
by (Step_tac 1); |
|
398 |
by (forw_inst_tac [("X","- x :: 'a set")] not_finite_UNIV_Compl 1); |
|
399 |
by (assume_tac 1); |
|
400 |
by (Step_tac 1 THEN Fast_tac 1); |
|
401 |
by (dres_inst_tac [("x","x")] bspec 1); |
|
402 |
by (Blast_tac 1); |
|
403 |
by (asm_full_simp_tac (simpset() addsimps [UNIV_diff_Compl]) 1); |
|
404 |
qed "FreeUltrafilter_contains_cofinite_set"; |
|
405 |
||
406 |
(*-------------------------------------------------------------------- |
|
407 |
We prove: 1. Existence of maximal filter i.e. ultrafilter |
|
408 |
2. Freeness property i.e ultrafilter is free |
|
409 |
Use a locale to prove various lemmas and then |
|
410 |
export main result: The Ultrafilter Theorem |
|
411 |
-------------------------------------------------------------------*) |
|
412 |
Open_locale "UFT"; |
|
413 |
||
414 |
Goalw [chain_def, thm "superfrechet_def", thm "frechet_def"] |
|
415 |
"!!(c :: 'a set set set). c : chain (superfrechet S) ==> Union c <= Pow S"; |
|
416 |
by (Step_tac 1); |
|
417 |
by (dtac subsetD 1 THEN assume_tac 1); |
|
418 |
by (Step_tac 1); |
|
419 |
by (dres_inst_tac [("X","X")] mem_FiltersetD3 1); |
|
420 |
by (Auto_tac); |
|
421 |
qed "chain_Un_subset_Pow"; |
|
422 |
||
423 |
Goalw [chain_def,Filter_def,is_Filter_def, |
|
424 |
thm "superfrechet_def", thm "frechet_def"] |
|
425 |
"!!(c :: 'a set set set). c: chain (superfrechet S) \ |
|
426 |
\ ==> !x: c. {} < x"; |
|
427 |
by (blast_tac (claset() addSIs [psubsetI]) 1); |
|
428 |
qed "mem_chain_psubset_empty"; |
|
429 |
||
430 |
Goal "!!(c :: 'a set set set). \ |
|
431 |
\ [| c: chain (superfrechet S);\ |
|
432 |
\ c ~= {} \ |
|
433 |
\ |]\ |
|
434 |
\ ==> Union(c) ~= {}"; |
|
435 |
by (dtac mem_chain_psubset_empty 1); |
|
436 |
by (Step_tac 1); |
|
437 |
by (dtac bspec 1 THEN assume_tac 1); |
|
438 |
by (auto_tac (claset() addDs [Union_upper,bspec], |
|
439 |
simpset() addsimps [psubset_def])); |
|
440 |
qed "chain_Un_not_empty"; |
|
441 |
||
442 |
Goalw [is_Filter_def,Filter_def,chain_def,thm "superfrechet_def"] |
|
443 |
"!!(c :: 'a set set set). \ |
|
444 |
\ c : chain (superfrechet S) \ |
|
445 |
\ ==> {} ~: Union(c)"; |
|
446 |
by (Blast_tac 1); |
|
447 |
qed "Filter_empty_not_mem_Un"; |
|
448 |
||
449 |
Goal "c: chain (superfrechet S) \ |
|
450 |
\ ==> ALL u : Union(c). ALL v: Union(c). u Int v : Union(c)"; |
|
451 |
by (Step_tac 1); |
|
452 |
by (forw_inst_tac [("x","X"),("y","Xa")] chainD 1); |
|
453 |
by (REPEAT(assume_tac 1)); |
|
454 |
by (dtac chainD2 1); |
|
455 |
by (etac disjE 1); |
|
456 |
by (res_inst_tac [("X","Xa")] UnionI 1 THEN assume_tac 1); |
|
457 |
by (dres_inst_tac [("A","X")] subsetD 1 THEN assume_tac 1); |
|
458 |
by (dres_inst_tac [("c","Xa")] subsetD 1 THEN assume_tac 1); |
|
459 |
by (res_inst_tac [("X","X")] UnionI 2 THEN assume_tac 2); |
|
460 |
by (dres_inst_tac [("A","Xa")] subsetD 2 THEN assume_tac 2); |
|
461 |
by (dres_inst_tac [("c","X")] subsetD 2 THEN assume_tac 2); |
|
462 |
by (auto_tac (claset() addIs [mem_FiltersetD1], |
|
463 |
simpset() addsimps [thm "superfrechet_def"])); |
|
464 |
qed "Filter_Un_Int"; |
|
465 |
||
466 |
Goal "c: chain (superfrechet S) \ |
|
467 |
\ ==> ALL u v. u: Union(c) & \ |
|
468 |
\ (u :: 'a set) <= v & v <= S --> v: Union(c)"; |
|
469 |
by (Step_tac 1); |
|
470 |
by (dtac chainD2 1); |
|
471 |
by (dtac subsetD 1 THEN assume_tac 1); |
|
472 |
by (rtac UnionI 1 THEN assume_tac 1); |
|
473 |
by (auto_tac (claset() addIs [mem_FiltersetD2], |
|
474 |
simpset() addsimps [thm "superfrechet_def"])); |
|
475 |
qed "Filter_Un_subset"; |
|
476 |
||
477 |
Goalw [chain_def,thm "superfrechet_def"] |
|
478 |
"!!(c :: 'a set set set). \ |
|
479 |
\ [| c: chain (superfrechet S);\ |
|
480 |
\ x: c \ |
|
481 |
\ |] ==> x : Filter S"; |
|
482 |
by (Blast_tac 1); |
|
483 |
qed "lemma_mem_chain_Filter"; |
|
484 |
||
485 |
Goalw [chain_def,thm "superfrechet_def"] |
|
486 |
"!!(c :: 'a set set set). \ |
|
487 |
\ [| c: chain (superfrechet S);\ |
|
488 |
\ x: c \ |
|
489 |
\ |] ==> frechet S <= x"; |
|
490 |
by (Blast_tac 1); |
|
491 |
qed "lemma_mem_chain_frechet_subset"; |
|
492 |
||
493 |
Goal "!!(c :: 'a set set set). \ |
|
494 |
\ [| c ~= {}; \ |
|
495 |
\ c : chain (superfrechet (UNIV :: 'a set))\ |
|
496 |
\ |] ==> Union c : superfrechet (UNIV)"; |
|
497 |
by (simp_tac (simpset() addsimps |
|
498 |
[thm "superfrechet_def",thm "frechet_def"]) 1); |
|
499 |
by (Step_tac 1); |
|
500 |
by (rtac mem_FiltersetI2 1); |
|
501 |
by (etac chain_Un_subset_Pow 1); |
|
502 |
by (rtac UnionI 1 THEN assume_tac 1); |
|
503 |
by (etac (lemma_mem_chain_Filter RS mem_FiltersetD4) 1 THEN assume_tac 1); |
|
504 |
by (etac chain_Un_not_empty 1); |
|
505 |
by (etac Filter_empty_not_mem_Un 2); |
|
506 |
by (etac Filter_Un_Int 2); |
|
507 |
by (etac Filter_Un_subset 2); |
|
508 |
by (subgoal_tac "xa : frechet (UNIV)" 2); |
|
509 |
by (rtac UnionI 2 THEN assume_tac 2); |
|
510 |
by (rtac (lemma_mem_chain_frechet_subset RS subsetD) 2); |
|
511 |
by (auto_tac (claset(),simpset() addsimps [thm "frechet_def"])); |
|
512 |
qed "Un_chain_mem_cofinite_Filter_set"; |
|
513 |
||
514 |
Goal "EX U: superfrechet (UNIV). \ |
|
515 |
\ ALL G: superfrechet (UNIV). U <= G --> U = G"; |
|
516 |
by (rtac Zorn_Lemma2 1); |
|
517 |
by (cut_facts_tac [thm "not_finite_UNIV" RS cofinite_Filter] 1); |
|
518 |
by (Step_tac 1); |
|
519 |
by (res_inst_tac [("Q","c={}")] (excluded_middle RS disjE) 1); |
|
520 |
by (res_inst_tac [("x","Union c")] bexI 1 THEN Blast_tac 1); |
|
521 |
by (rtac Un_chain_mem_cofinite_Filter_set 1 THEN REPEAT(assume_tac 1)); |
|
522 |
by (res_inst_tac [("x","frechet (UNIV)")] bexI 1 THEN Blast_tac 1); |
|
523 |
by (auto_tac (claset(), |
|
524 |
simpset() addsimps |
|
525 |
[thm "superfrechet_def", thm "frechet_def"])); |
|
526 |
qed "max_cofinite_Filter_Ex"; |
|
527 |
||
528 |
Goal "EX U: superfrechet UNIV. (\ |
|
529 |
\ ALL G: superfrechet UNIV. U <= G --> U = G) \ |
|
530 |
\ & (ALL x: U. ~finite x)"; |
|
531 |
by (cut_facts_tac [thm "not_finite_UNIV" RS |
|
532 |
(export max_cofinite_Filter_Ex)] 1); |
|
533 |
by (Step_tac 1); |
|
534 |
by (res_inst_tac [("x","U")] bexI 1); |
|
535 |
by (auto_tac (claset(),simpset() addsimps |
|
536 |
[thm "superfrechet_def", thm "frechet_def"])); |
|
537 |
by (dres_inst_tac [("c","- x")] subsetD 1); |
|
538 |
by (Asm_simp_tac 1); |
|
539 |
by (forw_inst_tac [("A","x"),("B","- x")] mem_FiltersetD1 1); |
|
540 |
by (dtac Filter_empty_not_mem 3); |
|
541 |
by (ALLGOALS(Asm_full_simp_tac )); |
|
542 |
qed "max_cofinite_Freefilter_Ex"; |
|
543 |
||
544 |
(*-------------------------------------------------------------------------------- |
|
545 |
There exists a free ultrafilter on any infinite set |
|
546 |
--------------------------------------------------------------------------------*) |
|
547 |
||
548 |
Goalw [FreeUltrafilter_def] "EX U. U: FreeUltrafilter (UNIV :: 'a set)"; |
|
549 |
by (cut_facts_tac [thm "not_finite_UNIV" RS (export max_cofinite_Freefilter_Ex)] 1); |
|
550 |
by (asm_full_simp_tac (simpset() addsimps |
|
551 |
[thm "superfrechet_def", Ultrafilter_iff, thm "frechet_def"]) 1); |
|
552 |
by (Step_tac 1); |
|
553 |
by (res_inst_tac [("x","U")] exI 1); |
|
554 |
by (Step_tac 1); |
|
555 |
by (Blast_tac 1); |
|
556 |
qed "FreeUltrafilter_ex"; |
|
557 |
||
558 |
bind_thm ("FreeUltrafilter_Ex", export FreeUltrafilter_ex); |
|
559 |
||
560 |
Close_locale "UFT"; |