author | wenzelm |
Tue, 21 Sep 1999 17:31:20 +0200 | |
changeset 7566 | c5a3f980a7af |
parent 7535 | 599d3414b51d |
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permissions | -rw-r--r-- |
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(* Title: HOL/Real/HahnBanach/Bounds.thy |
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ID: $Id$ |
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Author: Gertrud Bauer, TU Munich |
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*) |
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theory Bounds = Main + Real:; |
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section {* The sets of lower and upper bounds *}; |
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constdefs |
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is_LowerBound :: "('a::order) set => 'a set => 'a => bool" |
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"is_LowerBound A B == %x. x:A & (ALL y:B. x <= y)" |
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LowerBounds :: "('a::order) set => 'a set => 'a set" |
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"LowerBounds A B == Collect (is_LowerBound A B)" |
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is_UpperBound :: "('a::order) set => 'a set => 'a => bool" |
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"is_UpperBound A B == %x. x:A & (ALL y:B. y <= x)" |
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UpperBounds :: "('a::order) set => 'a set => 'a set" |
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"UpperBounds A B == Collect (is_UpperBound A B)"; |
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syntax |
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"_UPPERS" :: "[pttrn, 'a set, 'a => bool] => 'a set" ("(3UPPER'_BOUNDS _:_./ _)" 10) |
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"_UPPERS_U" :: "[pttrn, 'a => bool] => 'a set" ("(3UPPER'_BOUNDS _./ _)" 10) |
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"_LOWERS" :: "[pttrn, 'a set, 'a => bool] => 'a set" ("(3LOWER'_BOUNDS _:_./ _)" 10) |
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"_LOWERS_U" :: "[pttrn, 'a => bool] => 'a set" ("(3LOWER'_BOUNDS _./ _)" 10); |
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translations |
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"UPPER_BOUNDS x:A. P" == "UpperBounds A (Collect (%x. P))" |
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"UPPER_BOUNDS x. P" == "UPPER_BOUNDS x:UNIV. P" |
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"LOWER_BOUNDS x:A. P" == "LowerBounds A (Collect (%x. P))" |
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"LOWER_BOUNDS x. P" == "LOWER_BOUNDS x:UNIV. P"; |
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section {* Least and greatest elements *}; |
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constdefs |
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is_Least :: "('a::order) set => 'a => bool" |
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"is_Least B == is_LowerBound B B" |
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Least :: "('a::order) set => 'a" |
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"Least B == Eps (is_Least B)" |
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is_Greatest :: "('a::order) set => 'a => bool" |
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"is_Greatest B == is_UpperBound B B" |
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Greatest :: "('a::order) set => 'a" |
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"Greatest B == Eps (is_Greatest B)"; |
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syntax |
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"_LEAST" :: "[pttrn, 'a => bool] => 'a" ("(3LLEAST _./ _)" 10) |
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"_GREATEST" :: "[pttrn, 'a => bool] => 'a" ("(3GREATEST _./ _)" 10); |
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translations |
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"LLEAST x. P" == "Least {x. P}" |
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"GREATEST x. P" == "Greatest {x. P}"; |
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section {* Inf and Sup *}; |
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constdefs |
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is_Sup :: "('a::order) set => 'a set => 'a => bool" |
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"is_Sup A B x == isLub A B x" |
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Sup :: "('a::order) set => 'a set => 'a" |
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"Sup A B == Eps (is_Sup A B)" |
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is_Inf :: "('a::order) set => 'a set => 'a => bool" |
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"is_Inf A B x == x:A & is_Greatest (LowerBounds A B) x" |
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Inf :: "('a::order) set => 'a set => 'a" |
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"Inf A B == Eps (is_Inf A B)"; |
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syntax |
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"_SUP" :: "[pttrn, 'a set, 'a => bool] => 'a set" ("(3SUP _:_./ _)" 10) |
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"_SUP_U" :: "[pttrn, 'a => bool] => 'a set" ("(3SUP _./ _)" 10) |
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"_INF" :: "[pttrn, 'a set, 'a => bool] => 'a set" ("(3INF _:_./ _)" 10) |
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"_INF_U" :: "[pttrn, 'a => bool] => 'a set" ("(3INF _./ _)" 10); |
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translations |
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"SUP x:A. P" == "Sup A (Collect (%x. P))" |
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"SUP x. P" == "SUP x:UNIV. P" |
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"INF x:A. P" == "Inf A (Collect (%x. P))" |
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"INF x. P" == "INF x:UNIV. P"; |
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lemma ub_ge_sup: "isUb A B y ==> is_Sup A B s ==> s <= y"; |
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by (unfold is_Sup_def, rule isLub_le_isUb); |
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lemma sup_ub: "y:B ==> is_Sup A B s ==> y <= s"; |
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by (unfold is_Sup_def, rule isLubD2); |
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lemma sup_ub1: "ALL y:B. a <= y ==> is_Sup A B s ==> x:B ==> a <= s"; |
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proof -; |
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assume "ALL y:B. a <= y" "is_Sup A B s" "x:B"; |
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have "a <= x"; by (rule bspec); |
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also; have "x <= s"; by (rule sup_ub); |
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finally; show "a <= s"; .; |
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qed; |
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end; |