src/HOL/Algebra/Lattice.thy
author ballarin
Fri, 14 May 2004 19:29:22 +0200
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Change of theory hierarchy: Group is now based in Lattice.
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(*
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  Title:     HOL/Algebra/Lattice.thy
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  Id:        $Id$
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  Author:    Clemens Ballarin, started 7 November 2003
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  Copyright: Clemens Ballarin
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*)
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header {* Orders and Lattices *}
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theory Lattice = Main:
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text {* Object with a carrier set. *}
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record 'a partial_object =
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  carrier :: "'a set"
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subsection {* Partial Orders *}
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record 'a order = "'a partial_object" +
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  le :: "['a, 'a] => bool" (infixl "\<sqsubseteq>\<index>" 50)
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locale partial_order = struct L +
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  assumes refl [intro, simp]:
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                  "x \<in> carrier L ==> x \<sqsubseteq> x"
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    and anti_sym [intro]:
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                  "[| x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> carrier L; y \<in> carrier L |] ==> x = y"
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    and trans [trans]:
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                  "[| x \<sqsubseteq> y; y \<sqsubseteq> z;
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                   x \<in> carrier L; y \<in> carrier L; z \<in> carrier L |] ==> x \<sqsubseteq> z"
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constdefs (structure L)
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  less :: "[_, 'a, 'a] => bool" (infixl "\<sqsubset>\<index>" 50)
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  "x \<sqsubset> y == x \<sqsubseteq> y & x ~= y"
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  -- {* Upper and lower bounds of a set. *}
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  Upper :: "[_, 'a set] => 'a set"
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  "Upper L A == {u. (ALL x. x \<in> A \<inter> carrier L --> x \<sqsubseteq> u)} \<inter>
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                carrier L"
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  Lower :: "[_, 'a set] => 'a set"
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  "Lower L A == {l. (ALL x. x \<in> A \<inter> carrier L --> l \<sqsubseteq> x)} \<inter>
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                carrier L"
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  -- {* Least and greatest, as predicate. *}
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  least :: "[_, 'a, 'a set] => bool"
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  "least L l A == A \<subseteq> carrier L & l \<in> A & (ALL x : A. l \<sqsubseteq> x)"
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  greatest :: "[_, 'a, 'a set] => bool"
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  "greatest L g A == A \<subseteq> carrier L & g \<in> A & (ALL x : A. x \<sqsubseteq> g)"
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  -- {* Supremum and infimum *}
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  sup :: "[_, 'a set] => 'a" ("\<Squnion>\<index>_" [90] 90)
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  "\<Squnion>A == THE x. least L x (Upper L A)"
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  inf :: "[_, 'a set] => 'a" ("\<Sqinter>\<index>_" [90] 90)
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  "\<Sqinter>A == THE x. greatest L x (Lower L A)"
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  join :: "[_, 'a, 'a] => 'a" (infixl "\<squnion>\<index>" 65)
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  "x \<squnion> y == sup L {x, y}"
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  meet :: "[_, 'a, 'a] => 'a" (infixl "\<sqinter>\<index>" 65)
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  "x \<sqinter> y == inf L {x, y}"
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subsubsection {* Upper *}
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lemma Upper_closed [intro, simp]:
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  "Upper L A \<subseteq> carrier L"
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  by (unfold Upper_def) clarify
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lemma UpperD [dest]:
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  includes struct L
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  shows "[| u \<in> Upper L A; x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> u"
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  by (unfold Upper_def) blast
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lemma Upper_memI:
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  includes struct L
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  shows "[| !! y. y \<in> A ==> y \<sqsubseteq> x; x \<in> carrier L |] ==> x \<in> Upper L A"
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  by (unfold Upper_def) blast
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lemma Upper_antimono:
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  "A \<subseteq> B ==> Upper L B \<subseteq> Upper L A"
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  by (unfold Upper_def) blast
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subsubsection {* Lower *}
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lemma Lower_closed [intro, simp]:
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  "Lower L A \<subseteq> carrier L"
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  by (unfold Lower_def) clarify
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lemma LowerD [dest]:
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  includes struct L
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  shows "[| l \<in> Lower L A; x \<in> A; A \<subseteq> carrier L |] ==> l \<sqsubseteq> x"
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  by (unfold Lower_def) blast
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lemma Lower_memI:
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  includes struct L
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  shows "[| !! y. y \<in> A ==> x \<sqsubseteq> y; x \<in> carrier L |] ==> x \<in> Lower L A"
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  by (unfold Lower_def) blast
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lemma Lower_antimono:
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  "A \<subseteq> B ==> Lower L B \<subseteq> Lower L A"
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  by (unfold Lower_def) blast
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subsubsection {* least *}
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lemma least_carrier [intro, simp]:
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  shows "least L l A ==> l \<in> carrier L"
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  by (unfold least_def) fast
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lemma least_mem:
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  "least L l A ==> l \<in> A"
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  by (unfold least_def) fast
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lemma (in partial_order) least_unique:
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  "[| least L x A; least L y A |] ==> x = y"
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  by (unfold least_def) blast
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lemma least_le:
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  includes struct L
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  shows "[| least L x A; a \<in> A |] ==> x \<sqsubseteq> a"
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  by (unfold least_def) fast
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lemma least_UpperI:
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  includes struct L
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  assumes above: "!! x. x \<in> A ==> x \<sqsubseteq> s"
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    and below: "!! y. y \<in> Upper L A ==> s \<sqsubseteq> y"
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    and L: "A \<subseteq> carrier L"  "s \<in> carrier L"
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  shows "least L s (Upper L A)"
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proof -
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  have "Upper L A \<subseteq> carrier L" by simp
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  moreover from above L have "s \<in> Upper L A" by (simp add: Upper_def)
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  moreover from below have "ALL x : Upper L A. s \<sqsubseteq> x" by fast
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  ultimately show ?thesis by (simp add: least_def)
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qed
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subsubsection {* greatest *}
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lemma greatest_carrier [intro, simp]:
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  shows "greatest L l A ==> l \<in> carrier L"
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  by (unfold greatest_def) fast
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lemma greatest_mem:
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  "greatest L l A ==> l \<in> A"
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  by (unfold greatest_def) fast
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lemma (in partial_order) greatest_unique:
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  "[| greatest L x A; greatest L y A |] ==> x = y"
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  by (unfold greatest_def) blast
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lemma greatest_le:
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  includes struct L
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  shows "[| greatest L x A; a \<in> A |] ==> a \<sqsubseteq> x"
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  by (unfold greatest_def) fast
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lemma greatest_LowerI:
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  includes struct L
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  assumes below: "!! x. x \<in> A ==> i \<sqsubseteq> x"
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    and above: "!! y. y \<in> Lower L A ==> y \<sqsubseteq> i"
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    and L: "A \<subseteq> carrier L"  "i \<in> carrier L"
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  shows "greatest L i (Lower L A)"
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proof -
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  have "Lower L A \<subseteq> carrier L" by simp
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  moreover from below L have "i \<in> Lower L A" by (simp add: Lower_def)
4deda204e1d8 improved syntax;
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  moreover from above have "ALL x : Lower L A. x \<sqsubseteq> i" by fast
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  ultimately show ?thesis by (simp add: greatest_def)
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qed
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   171
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subsection {* Lattices *}
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   174
2cb6ff394bfb Various changes to HOL-Algebra;
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locale lattice = partial_order +
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  assumes sup_of_two_exists:
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    "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. least L s (Upper L {x, y})"
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   178
    and inf_of_two_exists:
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   179
    "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. greatest L s (Lower L {x, y})"
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   180
2cb6ff394bfb Various changes to HOL-Algebra;
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   181
lemma least_Upper_above:
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  includes struct L
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  shows "[| least L s (Upper L A); x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> s"
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parents:
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   184
  by (unfold least_def) blast
2cb6ff394bfb Various changes to HOL-Algebra;
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parents:
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   185
2cb6ff394bfb Various changes to HOL-Algebra;
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   186
lemma greatest_Lower_above:
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  includes struct L
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  shows "[| greatest L i (Lower L A); x \<in> A; A \<subseteq> carrier L |] ==> i \<sqsubseteq> x"
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parents:
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   189
  by (unfold greatest_def) blast
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   190
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subsubsection {* Supremum *}
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   193
2cb6ff394bfb Various changes to HOL-Algebra;
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   194
lemma (in lattice) joinI:
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  "[| !!l. least L l (Upper L {x, y}) ==> P l; x \<in> carrier L; y \<in> carrier L |]
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   196
  ==> P (x \<squnion> y)"
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   197
proof (unfold join_def sup_def)
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  assume L: "x \<in> carrier L"  "y \<in> carrier L"
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   199
    and P: "!!l. least L l (Upper L {x, y}) ==> P l"
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   200
  with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
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   201
  with L show "P (THE l. least L l (Upper L {x, y}))"
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    by (fast intro: theI2 least_unique P)
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   203
qed
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   204
2cb6ff394bfb Various changes to HOL-Algebra;
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   205
lemma (in lattice) join_closed [simp]:
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   206
  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<squnion> y \<in> carrier L"
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   207
  by (rule joinI) (rule least_carrier)
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   208
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   209
lemma (in partial_order) sup_of_singletonI:      (* only reflexivity needed ? *)
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   210
  "x \<in> carrier L ==> least L x (Upper L {x})"
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   211
  by (rule least_UpperI) fast+
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ballarin
parents:
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   212
2cb6ff394bfb Various changes to HOL-Algebra;
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   213
lemma (in partial_order) sup_of_singleton [simp]:
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   214
  includes struct L
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   215
  shows "x \<in> carrier L ==> \<Squnion>{x} = x"
14551
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parents:
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   216
  by (unfold sup_def) (blast intro: least_unique least_UpperI sup_of_singletonI)
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   217
14666
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   218
65f8680c3f16 improved notation;
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   219
text {* Condition on @{text A}: supremum exists. *}
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   220
2cb6ff394bfb Various changes to HOL-Algebra;
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   221
lemma (in lattice) sup_insertI:
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   222
  "[| !!s. least L s (Upper L (insert x A)) ==> P s;
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   223
  least L a (Upper L A); x \<in> carrier L; A \<subseteq> carrier L |]
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   224
  ==> P (\<Squnion>(insert x A))"
14551
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   225
proof (unfold sup_def)
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   226
  assume L: "x \<in> carrier L"  "A \<subseteq> carrier L"
14551
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   227
    and P: "!!l. least L l (Upper L (insert x A)) ==> P l"
2cb6ff394bfb Various changes to HOL-Algebra;
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parents:
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   228
    and least_a: "least L a (Upper L A)"
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parents:
diff changeset
   229
  from L least_a have La: "a \<in> carrier L" by simp
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parents:
diff changeset
   230
  from L sup_of_two_exists least_a
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ballarin
parents:
diff changeset
   231
  obtain s where least_s: "least L s (Upper L {a, x})" by blast
2cb6ff394bfb Various changes to HOL-Algebra;
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parents:
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   232
  show "P (THE l. least L l (Upper L (insert x A)))"
14693
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   233
  proof (rule theI2)
14551
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parents:
diff changeset
   234
    show "least L s (Upper L (insert x A))"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   235
    proof (rule least_UpperI)
2cb6ff394bfb Various changes to HOL-Algebra;
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parents:
diff changeset
   236
      fix z
14693
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   237
      assume "z \<in> insert x A"
4deda204e1d8 improved syntax;
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parents: 14666
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   238
      then show "z \<sqsubseteq> s"
4deda204e1d8 improved syntax;
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diff changeset
   239
      proof
4deda204e1d8 improved syntax;
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   240
        assume "z = x" then show ?thesis
4deda204e1d8 improved syntax;
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   241
          by (simp add: least_Upper_above [OF least_s] L La)
4deda204e1d8 improved syntax;
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   242
      next
4deda204e1d8 improved syntax;
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diff changeset
   243
        assume "z \<in> A"
4deda204e1d8 improved syntax;
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   244
        with L least_s least_a show ?thesis
4deda204e1d8 improved syntax;
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diff changeset
   245
          by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
4deda204e1d8 improved syntax;
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   246
      qed
4deda204e1d8 improved syntax;
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   247
    next
4deda204e1d8 improved syntax;
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   248
      fix y
4deda204e1d8 improved syntax;
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   249
      assume y: "y \<in> Upper L (insert x A)"
4deda204e1d8 improved syntax;
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diff changeset
   250
      show "s \<sqsubseteq> y"
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   251
      proof (rule least_le [OF least_s], rule Upper_memI)
4deda204e1d8 improved syntax;
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parents: 14666
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   252
	fix z
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   253
	assume z: "z \<in> {a, x}"
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   254
	then show "z \<sqsubseteq> y"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   255
	proof
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   256
          have y': "y \<in> Upper L A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   257
            apply (rule subsetD [where A = "Upper L (insert x A)"])
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   258
            apply (rule Upper_antimono) apply clarify apply assumption
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   259
            done
4deda204e1d8 improved syntax;
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diff changeset
   260
          assume "z = a"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   261
          with y' least_a show ?thesis by (fast dest: least_le)
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   262
	next
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   263
	  assume "z \<in> {x}"  (* FIXME "z = x"; declare specific elim rule for "insert x {}" (!?) *)
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   264
          with y L show ?thesis by blast
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   265
	qed
4deda204e1d8 improved syntax;
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diff changeset
   266
      qed (rule Upper_closed [THEN subsetD])
4deda204e1d8 improved syntax;
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diff changeset
   267
    next
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   268
      from L show "insert x A \<subseteq> carrier L" by simp
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   269
      from least_s show "s \<in> carrier L" by simp
14551
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ballarin
parents:
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   270
    qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   271
  next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   272
    fix l
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   273
    assume least_l: "least L l (Upper L (insert x A))"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   274
    show "l = s"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   275
    proof (rule least_unique)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   276
      show "least L s (Upper L (insert x A))"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   277
      proof (rule least_UpperI)
14693
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   278
        fix z
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   279
        assume "z \<in> insert x A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   280
        then show "z \<sqsubseteq> s"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   281
	proof
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   282
          assume "z = x" then show ?thesis
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   283
            by (simp add: least_Upper_above [OF least_s] L La)
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   284
	next
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   285
          assume "z \<in> A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   286
          with L least_s least_a show ?thesis
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   287
            by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
14551
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ballarin
parents:
diff changeset
   288
	qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   289
      next
14693
4deda204e1d8 improved syntax;
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diff changeset
   290
        fix y
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   291
        assume y: "y \<in> Upper L (insert x A)"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   292
        show "s \<sqsubseteq> y"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   293
        proof (rule least_le [OF least_s], rule Upper_memI)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   294
          fix z
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   295
          assume z: "z \<in> {a, x}"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   296
          then show "z \<sqsubseteq> y"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   297
          proof
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   298
            have y': "y \<in> Upper L A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   299
	      apply (rule subsetD [where A = "Upper L (insert x A)"])
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   300
	      apply (rule Upper_antimono) apply clarify apply assumption
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   301
	      done
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   302
            assume "z = a"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   303
            with y' least_a show ?thesis by (fast dest: least_le)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   304
	  next
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   305
            assume "z \<in> {x}"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   306
            with y L show ?thesis by blast
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   307
          qed
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   308
        qed (rule Upper_closed [THEN subsetD])
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   309
      next
14693
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   310
        from L show "insert x A \<subseteq> carrier L" by simp
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   311
        from least_s show "s \<in> carrier L" by simp
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   312
      qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   313
    qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   314
  qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   315
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   316
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   317
lemma (in lattice) finite_sup_least:
14693
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   318
  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> least L (\<Squnion>A) (Upper L A)"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   319
proof (induct set: Finites)
14693
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   320
  case empty
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   321
  then show ?case by simp
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   322
next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   323
  case (insert A x)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   324
  show ?case
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   325
  proof (cases "A = {}")
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   326
    case True
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   327
    with insert show ?thesis by (simp add: sup_of_singletonI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   328
  next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   329
    case False
14693
4deda204e1d8 improved syntax;
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parents: 14666
diff changeset
   330
    with insert have "least L (\<Squnion>A) (Upper L A)" by simp
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   331
    with _ show ?thesis
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   332
      by (rule sup_insertI) (simp_all add: insert [simplified])
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   333
  qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   334
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   335
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   336
lemma (in lattice) finite_sup_insertI:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   337
  assumes P: "!!l. least L l (Upper L (insert x A)) ==> P l"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   338
    and xA: "finite A"  "x \<in> carrier L"  "A \<subseteq> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   339
  shows "P (\<Squnion> (insert x A))"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   340
proof (cases "A = {}")
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   341
  case True with P and xA show ?thesis
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   342
    by (simp add: sup_of_singletonI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   343
next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   344
  case False with P and xA show ?thesis
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   345
    by (simp add: sup_insertI finite_sup_least)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   346
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   347
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   348
lemma (in lattice) finite_sup_closed:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   349
  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Squnion>A \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   350
proof (induct set: Finites)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   351
  case empty then show ?case by simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   352
next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   353
  case (insert A x) then show ?case
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   354
    by - (rule finite_sup_insertI, simp_all)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   355
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   356
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   357
lemma (in lattice) join_left:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   358
  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> x \<squnion> y"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   359
  by (rule joinI [folded join_def]) (blast dest: least_mem)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   360
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   361
lemma (in lattice) join_right:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   362
  "[| x \<in> carrier L; y \<in> carrier L |] ==> y \<sqsubseteq> x \<squnion> y"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   363
  by (rule joinI [folded join_def]) (blast dest: least_mem)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   364
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   365
lemma (in lattice) sup_of_two_least:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   366
  "[| x \<in> carrier L; y \<in> carrier L |] ==> least L (\<Squnion>{x, y}) (Upper L {x, y})"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   367
proof (unfold sup_def)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   368
  assume L: "x \<in> carrier L"  "y \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   369
  with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   370
  with L show "least L (THE xa. least L xa (Upper L {x, y})) (Upper L {x, y})"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   371
  by (fast intro: theI2 least_unique)  (* blast fails *)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   372
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   373
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   374
lemma (in lattice) join_le:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   375
  assumes sub: "x \<sqsubseteq> z"  "y \<sqsubseteq> z"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   376
    and L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   377
  shows "x \<squnion> y \<sqsubseteq> z"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   378
proof (rule joinI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   379
  fix s
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   380
  assume "least L s (Upper L {x, y})"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   381
  with sub L show "s \<sqsubseteq> z" by (fast elim: least_le intro: Upper_memI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   382
qed
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   383
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   384
lemma (in lattice) join_assoc_lemma:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   385
  assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   386
  shows "x \<squnion> (y \<squnion> z) = \<Squnion>{x, y, z}"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   387
proof (rule finite_sup_insertI)
14651
02b8f3bcf7fe improved notation;
wenzelm
parents: 14577
diff changeset
   388
  -- {* The textbook argument in Jacobson I, p 457 *}
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   389
  fix s
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   390
  assume sup: "least L s (Upper L {x, y, z})"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   391
  show "x \<squnion> (y \<squnion> z) = s"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   392
  proof (rule anti_sym)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   393
    from sup L show "x \<squnion> (y \<squnion> z) \<sqsubseteq> s"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   394
      by (fastsimp intro!: join_le elim: least_Upper_above)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   395
  next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   396
    from sup L show "s \<sqsubseteq> x \<squnion> (y \<squnion> z)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   397
    by (erule_tac least_le)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   398
      (blast intro!: Upper_memI intro: trans join_left join_right join_closed)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   399
  qed (simp_all add: L least_carrier [OF sup])
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   400
qed (simp_all add: L)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   401
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   402
lemma join_comm:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   403
  includes struct L
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   404
  shows "x \<squnion> y = y \<squnion> x"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   405
  by (unfold join_def) (simp add: insert_commute)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   406
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   407
lemma (in lattice) join_assoc:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   408
  assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   409
  shows "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   410
proof -
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   411
  have "(x \<squnion> y) \<squnion> z = z \<squnion> (x \<squnion> y)" by (simp only: join_comm)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   412
  also from L have "... = \<Squnion>{z, x, y}" by (simp add: join_assoc_lemma)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   413
  also from L have "... = \<Squnion>{x, y, z}" by (simp add: insert_commute)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   414
  also from L have "... = x \<squnion> (y \<squnion> z)" by (simp add: join_assoc_lemma)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   415
  finally show ?thesis .
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   416
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   417
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   418
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   419
subsubsection {* Infimum *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   420
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   421
lemma (in lattice) meetI:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   422
  "[| !!i. greatest L i (Lower L {x, y}) ==> P i;
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   423
  x \<in> carrier L; y \<in> carrier L |]
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   424
  ==> P (x \<sqinter> y)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   425
proof (unfold meet_def inf_def)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   426
  assume L: "x \<in> carrier L"  "y \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   427
    and P: "!!g. greatest L g (Lower L {x, y}) ==> P g"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   428
  with inf_of_two_exists obtain i where "greatest L i (Lower L {x, y})" by fast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   429
  with L show "P (THE g. greatest L g (Lower L {x, y}))"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   430
  by (fast intro: theI2 greatest_unique P)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   431
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   432
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   433
lemma (in lattice) meet_closed [simp]:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   434
  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<in> carrier L"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   435
  by (rule meetI) (rule greatest_carrier)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   436
14651
02b8f3bcf7fe improved notation;
wenzelm
parents: 14577
diff changeset
   437
lemma (in partial_order) inf_of_singletonI:      (* only reflexivity needed ? *)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   438
  "x \<in> carrier L ==> greatest L x (Lower L {x})"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   439
  by (rule greatest_LowerI) fast+
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   440
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   441
lemma (in partial_order) inf_of_singleton [simp]:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   442
  includes struct L
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   443
  shows "x \<in> carrier L ==> \<Sqinter> {x} = x"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   444
  by (unfold inf_def) (blast intro: greatest_unique greatest_LowerI inf_of_singletonI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   445
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   446
text {* Condition on A: infimum exists. *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   447
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   448
lemma (in lattice) inf_insertI:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   449
  "[| !!i. greatest L i (Lower L (insert x A)) ==> P i;
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   450
  greatest L a (Lower L A); x \<in> carrier L; A \<subseteq> carrier L |]
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   451
  ==> P (\<Sqinter>(insert x A))"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   452
proof (unfold inf_def)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   453
  assume L: "x \<in> carrier L"  "A \<subseteq> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   454
    and P: "!!g. greatest L g (Lower L (insert x A)) ==> P g"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   455
    and greatest_a: "greatest L a (Lower L A)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   456
  from L greatest_a have La: "a \<in> carrier L" by simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   457
  from L inf_of_two_exists greatest_a
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   458
  obtain i where greatest_i: "greatest L i (Lower L {a, x})" by blast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   459
  show "P (THE g. greatest L g (Lower L (insert x A)))"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   460
  proof (rule theI2)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   461
    show "greatest L i (Lower L (insert x A))"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   462
    proof (rule greatest_LowerI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   463
      fix z
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   464
      assume "z \<in> insert x A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   465
      then show "i \<sqsubseteq> z"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   466
      proof
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   467
        assume "z = x" then show ?thesis
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   468
          by (simp add: greatest_Lower_above [OF greatest_i] L La)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   469
      next
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   470
        assume "z \<in> A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   471
        with L greatest_i greatest_a show ?thesis
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   472
          by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   473
      qed
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   474
    next
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   475
      fix y
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   476
      assume y: "y \<in> Lower L (insert x A)"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   477
      show "y \<sqsubseteq> i"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   478
      proof (rule greatest_le [OF greatest_i], rule Lower_memI)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   479
	fix z
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   480
	assume z: "z \<in> {a, x}"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   481
	then show "y \<sqsubseteq> z"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   482
	proof
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   483
          have y': "y \<in> Lower L A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   484
            apply (rule subsetD [where A = "Lower L (insert x A)"])
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   485
            apply (rule Lower_antimono) apply clarify apply assumption
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   486
            done
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   487
          assume "z = a"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   488
          with y' greatest_a show ?thesis by (fast dest: greatest_le)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   489
	next
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   490
          assume "z \<in> {x}"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   491
          with y L show ?thesis by blast
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   492
	qed
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   493
      qed (rule Lower_closed [THEN subsetD])
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   494
    next
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   495
      from L show "insert x A \<subseteq> carrier L" by simp
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   496
      from greatest_i show "i \<in> carrier L" by simp
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   497
    qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   498
  next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   499
    fix g
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   500
    assume greatest_g: "greatest L g (Lower L (insert x A))"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   501
    show "g = i"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   502
    proof (rule greatest_unique)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   503
      show "greatest L i (Lower L (insert x A))"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   504
      proof (rule greatest_LowerI)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   505
        fix z
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   506
        assume "z \<in> insert x A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   507
        then show "i \<sqsubseteq> z"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   508
	proof
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   509
          assume "z = x" then show ?thesis
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   510
            by (simp add: greatest_Lower_above [OF greatest_i] L La)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   511
	next
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   512
          assume "z \<in> A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   513
          with L greatest_i greatest_a show ?thesis
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   514
            by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   515
        qed
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   516
      next
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   517
        fix y
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   518
        assume y: "y \<in> Lower L (insert x A)"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   519
        show "y \<sqsubseteq> i"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   520
        proof (rule greatest_le [OF greatest_i], rule Lower_memI)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   521
          fix z
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   522
          assume z: "z \<in> {a, x}"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   523
          then show "y \<sqsubseteq> z"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   524
          proof
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   525
            have y': "y \<in> Lower L A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   526
	      apply (rule subsetD [where A = "Lower L (insert x A)"])
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   527
	      apply (rule Lower_antimono) apply clarify apply assumption
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   528
	      done
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   529
            assume "z = a"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   530
            with y' greatest_a show ?thesis by (fast dest: greatest_le)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   531
	  next
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   532
            assume "z \<in> {x}"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   533
            with y L show ?thesis by blast
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   534
	  qed
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   535
        qed (rule Lower_closed [THEN subsetD])
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   536
      next
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   537
        from L show "insert x A \<subseteq> carrier L" by simp
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   538
        from greatest_i show "i \<in> carrier L" by simp
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   539
      qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   540
    qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   541
  qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   542
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   543
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   544
lemma (in lattice) finite_inf_greatest:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   545
  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> greatest L (\<Sqinter>A) (Lower L A)"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   546
proof (induct set: Finites)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   547
  case empty then show ?case by simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   548
next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   549
  case (insert A x)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   550
  show ?case
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   551
  proof (cases "A = {}")
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   552
    case True
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   553
    with insert show ?thesis by (simp add: inf_of_singletonI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   554
  next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   555
    case False
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   556
    from insert show ?thesis
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   557
    proof (rule_tac inf_insertI)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   558
      from False insert show "greatest L (\<Sqinter>A) (Lower L A)" by simp
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   559
    qed simp_all
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   560
  qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   561
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   562
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   563
lemma (in lattice) finite_inf_insertI:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   564
  assumes P: "!!i. greatest L i (Lower L (insert x A)) ==> P i"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   565
    and xA: "finite A"  "x \<in> carrier L"  "A \<subseteq> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   566
  shows "P (\<Sqinter> (insert x A))"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   567
proof (cases "A = {}")
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   568
  case True with P and xA show ?thesis
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   569
    by (simp add: inf_of_singletonI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   570
next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   571
  case False with P and xA show ?thesis
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   572
    by (simp add: inf_insertI finite_inf_greatest)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   573
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   574
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   575
lemma (in lattice) finite_inf_closed:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   576
  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Sqinter>A \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   577
proof (induct set: Finites)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   578
  case empty then show ?case by simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   579
next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   580
  case (insert A x) then show ?case
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   581
    by (rule_tac finite_inf_insertI) (simp_all)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   582
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   583
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   584
lemma (in lattice) meet_left:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   585
  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> x"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   586
  by (rule meetI [folded meet_def]) (blast dest: greatest_mem)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   587
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   588
lemma (in lattice) meet_right:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   589
  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> y"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   590
  by (rule meetI [folded meet_def]) (blast dest: greatest_mem)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   591
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   592
lemma (in lattice) inf_of_two_greatest:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   593
  "[| x \<in> carrier L; y \<in> carrier L |] ==>
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   594
  greatest L (\<Sqinter> {x, y}) (Lower L {x, y})"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   595
proof (unfold inf_def)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   596
  assume L: "x \<in> carrier L"  "y \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   597
  with inf_of_two_exists obtain s where "greatest L s (Lower L {x, y})" by fast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   598
  with L
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   599
  show "greatest L (THE xa. greatest L xa (Lower L {x, y})) (Lower L {x, y})"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   600
  by (fast intro: theI2 greatest_unique)  (* blast fails *)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   601
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   602
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   603
lemma (in lattice) meet_le:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   604
  assumes sub: "z \<sqsubseteq> x"  "z \<sqsubseteq> y"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   605
    and L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   606
  shows "z \<sqsubseteq> x \<sqinter> y"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   607
proof (rule meetI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   608
  fix i
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   609
  assume "greatest L i (Lower L {x, y})"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   610
  with sub L show "z \<sqsubseteq> i" by (fast elim: greatest_le intro: Lower_memI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   611
qed
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   612
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   613
lemma (in lattice) meet_assoc_lemma:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   614
  assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   615
  shows "x \<sqinter> (y \<sqinter> z) = \<Sqinter>{x, y, z}"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   616
proof (rule finite_inf_insertI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   617
  txt {* The textbook argument in Jacobson I, p 457 *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   618
  fix i
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   619
  assume inf: "greatest L i (Lower L {x, y, z})"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   620
  show "x \<sqinter> (y \<sqinter> z) = i"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   621
  proof (rule anti_sym)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   622
    from inf L show "i \<sqsubseteq> x \<sqinter> (y \<sqinter> z)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   623
      by (fastsimp intro!: meet_le elim: greatest_Lower_above)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   624
  next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   625
    from inf L show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> i"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   626
    by (erule_tac greatest_le)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   627
      (blast intro!: Lower_memI intro: trans meet_left meet_right meet_closed)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   628
  qed (simp_all add: L greatest_carrier [OF inf])
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   629
qed (simp_all add: L)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   630
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   631
lemma meet_comm:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   632
  includes struct L
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   633
  shows "x \<sqinter> y = y \<sqinter> x"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   634
  by (unfold meet_def) (simp add: insert_commute)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   635
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   636
lemma (in lattice) meet_assoc:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   637
  assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   638
  shows "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   639
proof -
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   640
  have "(x \<sqinter> y) \<sqinter> z = z \<sqinter> (x \<sqinter> y)" by (simp only: meet_comm)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   641
  also from L have "... = \<Sqinter> {z, x, y}" by (simp add: meet_assoc_lemma)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   642
  also from L have "... = \<Sqinter> {x, y, z}" by (simp add: insert_commute)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   643
  also from L have "... = x \<sqinter> (y \<sqinter> z)" by (simp add: meet_assoc_lemma)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   644
  finally show ?thesis .
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   645
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   646
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   647
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   648
subsection {* Total Orders *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   649
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   650
locale total_order = lattice +
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   651
  assumes total: "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   652
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   653
text {* Introduction rule: the usual definition of total order *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   654
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   655
lemma (in partial_order) total_orderI:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   656
  assumes total: "!!x y. [| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   657
  shows "total_order L"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   658
proof (rule total_order.intro)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   659
  show "lattice_axioms L"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   660
  proof (rule lattice_axioms.intro)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   661
    fix x y
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   662
    assume L: "x \<in> carrier L"  "y \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   663
    show "EX s. least L s (Upper L {x, y})"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   664
    proof -
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   665
      note total L
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   666
      moreover
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   667
      {
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   668
        assume "x \<sqsubseteq> y"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   669
        with L have "least L y (Upper L {x, y})"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   670
          by (rule_tac least_UpperI) auto
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   671
      }
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   672
      moreover
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   673
      {
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   674
        assume "y \<sqsubseteq> x"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   675
        with L have "least L x (Upper L {x, y})"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   676
          by (rule_tac least_UpperI) auto
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   677
      }
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   678
      ultimately show ?thesis by blast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   679
    qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   680
  next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   681
    fix x y
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   682
    assume L: "x \<in> carrier L"  "y \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   683
    show "EX i. greatest L i (Lower L {x, y})"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   684
    proof -
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   685
      note total L
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   686
      moreover
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   687
      {
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   688
        assume "y \<sqsubseteq> x"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   689
        with L have "greatest L y (Lower L {x, y})"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   690
          by (rule_tac greatest_LowerI) auto
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   691
      }
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   692
      moreover
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   693
      {
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   694
        assume "x \<sqsubseteq> y"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   695
        with L have "greatest L x (Lower L {x, y})"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   696
          by (rule_tac greatest_LowerI) auto
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   697
      }
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   698
      ultimately show ?thesis by blast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   699
    qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   700
  qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   701
qed (assumption | rule total_order_axioms.intro)+
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   702
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   703
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   704
subsection {* Complete lattices *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   705
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   706
locale complete_lattice = lattice +
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   707
  assumes sup_exists:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   708
    "[| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   709
    and inf_exists:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   710
    "[| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   711
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   712
text {* Introduction rule: the usual definition of complete lattice *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   713
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   714
lemma (in partial_order) complete_latticeI:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   715
  assumes sup_exists:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   716
    "!!A. [| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   717
    and inf_exists:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   718
    "!!A. [| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   719
  shows "complete_lattice L"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   720
proof (rule complete_lattice.intro)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   721
  show "lattice_axioms L"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   722
    by (rule lattice_axioms.intro) (blast intro: sup_exists inf_exists)+
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   723
qed (assumption | rule complete_lattice_axioms.intro)+
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   724
14651
02b8f3bcf7fe improved notation;
wenzelm
parents: 14577
diff changeset
   725
constdefs (structure L)
02b8f3bcf7fe improved notation;
wenzelm
parents: 14577
diff changeset
   726
  top :: "_ => 'a" ("\<top>\<index>")
02b8f3bcf7fe improved notation;
wenzelm
parents: 14577
diff changeset
   727
  "\<top> == sup L (carrier L)"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   728
14651
02b8f3bcf7fe improved notation;
wenzelm
parents: 14577
diff changeset
   729
  bottom :: "_ => 'a" ("\<bottom>\<index>")
02b8f3bcf7fe improved notation;
wenzelm
parents: 14577
diff changeset
   730
  "\<bottom> == inf L (carrier L)"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   731
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   732
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   733
lemma (in complete_lattice) supI:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   734
  "[| !!l. least L l (Upper L A) ==> P l; A \<subseteq> carrier L |]
14651
02b8f3bcf7fe improved notation;
wenzelm
parents: 14577
diff changeset
   735
  ==> P (\<Squnion>A)"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   736
proof (unfold sup_def)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   737
  assume L: "A \<subseteq> carrier L"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   738
    and P: "!!l. least L l (Upper L A) ==> P l"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   739
  with sup_exists obtain s where "least L s (Upper L A)" by blast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   740
  with L show "P (THE l. least L l (Upper L A))"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   741
  by (fast intro: theI2 least_unique P)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   742
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   743
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   744
lemma (in complete_lattice) sup_closed [simp]:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   745
  "A \<subseteq> carrier L ==> \<Squnion>A \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   746
  by (rule supI) simp_all
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   747
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   748
lemma (in complete_lattice) top_closed [simp, intro]:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   749
  "\<top> \<in> carrier L"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   750
  by (unfold top_def) simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   751
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   752
lemma (in complete_lattice) infI:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   753
  "[| !!i. greatest L i (Lower L A) ==> P i; A \<subseteq> carrier L |]
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   754
  ==> P (\<Sqinter>A)"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   755
proof (unfold inf_def)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   756
  assume L: "A \<subseteq> carrier L"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   757
    and P: "!!l. greatest L l (Lower L A) ==> P l"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   758
  with inf_exists obtain s where "greatest L s (Lower L A)" by blast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   759
  with L show "P (THE l. greatest L l (Lower L A))"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   760
  by (fast intro: theI2 greatest_unique P)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   761
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   762
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   763
lemma (in complete_lattice) inf_closed [simp]:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   764
  "A \<subseteq> carrier L ==> \<Sqinter>A \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   765
  by (rule infI) simp_all
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   766
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   767
lemma (in complete_lattice) bottom_closed [simp, intro]:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   768
  "\<bottom> \<in> carrier L"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   769
  by (unfold bottom_def) simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   770
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   771
text {* Jacobson: Theorem 8.1 *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   772
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   773
lemma Lower_empty [simp]:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   774
  "Lower L {} = carrier L"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   775
  by (unfold Lower_def) simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   776
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   777
lemma Upper_empty [simp]:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   778
  "Upper L {} = carrier L"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   779
  by (unfold Upper_def) simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   780
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   781
theorem (in partial_order) complete_lattice_criterion1:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   782
  assumes top_exists: "EX g. greatest L g (carrier L)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   783
    and inf_exists:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   784
      "!!A. [| A \<subseteq> carrier L; A ~= {} |] ==> EX i. greatest L i (Lower L A)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   785
  shows "complete_lattice L"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   786
proof (rule complete_latticeI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   787
  from top_exists obtain top where top: "greatest L top (carrier L)" ..
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   788
  fix A
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   789
  assume L: "A \<subseteq> carrier L"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   790
  let ?B = "Upper L A"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   791
  from L top have "top \<in> ?B" by (fast intro!: Upper_memI intro: greatest_le)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   792
  then have B_non_empty: "?B ~= {}" by fast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   793
  have B_L: "?B \<subseteq> carrier L" by simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   794
  from inf_exists [OF B_L B_non_empty]
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   795
  obtain b where b_inf_B: "greatest L b (Lower L ?B)" ..
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   796
  have "least L b (Upper L A)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   797
apply (rule least_UpperI)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   798
   apply (rule greatest_le [where A = "Lower L ?B"])
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   799
    apply (rule b_inf_B)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   800
   apply (rule Lower_memI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   801
    apply (erule UpperD)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   802
     apply assumption
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   803
    apply (rule L)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   804
   apply (fast intro: L [THEN subsetD])
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   805
  apply (erule greatest_Lower_above [OF b_inf_B])
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   806
  apply simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   807
 apply (rule L)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   808
apply (rule greatest_carrier [OF b_inf_B]) (* rename rule: _closed *)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   809
done
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   810
  then show "EX s. least L s (Upper L A)" ..
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   811
next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   812
  fix A
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   813
  assume L: "A \<subseteq> carrier L"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   814
  show "EX i. greatest L i (Lower L A)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   815
  proof (cases "A = {}")
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   816
    case True then show ?thesis
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   817
      by (simp add: top_exists)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   818
  next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   819
    case False with L show ?thesis
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   820
      by (rule inf_exists)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   821
  qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   822
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   823
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   824
(* TODO: prove dual version *)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   825
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   826
subsection {* Examples *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   827
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   828
subsubsection {* Powerset of a set is a complete lattice *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   829
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   830
theorem powerset_is_complete_lattice:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   831
  "complete_lattice (| carrier = Pow A, le = op \<subseteq> |)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   832
  (is "complete_lattice ?L")
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   833
proof (rule partial_order.complete_latticeI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   834
  show "partial_order ?L"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   835
    by (rule partial_order.intro) auto
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   836
next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   837
  fix B
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   838
  assume "B \<subseteq> carrier ?L"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   839
  then have "least ?L (\<Union> B) (Upper ?L B)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   840
    by (fastsimp intro!: least_UpperI simp: Upper_def)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   841
  then show "EX s. least ?L s (Upper ?L B)" ..
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   842
next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   843
  fix B
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   844
  assume "B \<subseteq> carrier ?L"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   845
  then have "greatest ?L (\<Inter> B \<inter> A) (Lower ?L B)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   846
    txt {* @{term "\<Inter> B"} is not the infimum of @{term B}:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   847
      @{term "\<Inter> {} = UNIV"} which is in general bigger than @{term "A"}! *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   848
    by (fastsimp intro!: greatest_LowerI simp: Lower_def)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   849
  then show "EX i. greatest ?L i (Lower ?L B)" ..
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   850
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   851
14751
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   852
text {* An other example, that of the lattice of subgroups of a group,
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   853
  can be found in Group theory (Section~\ref{sec:subgroup-lattice}). *}
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   854
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   855
end