src/HOL/Algebra/Lattice.thy
author wenzelm
Thu Apr 22 11:01:34 2004 +0200 (2004-04-22)
changeset 14651 02b8f3bcf7fe
parent 14577 dbb95b825244
child 14666 65f8680c3f16
permissions -rw-r--r--
improved notation;
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(*
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  Title:     Orders and Lattices
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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 {* Order and Lattices *}
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theory Lattice = Group:
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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 order_syntax = struct L
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locale partial_order = order_syntax +
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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 --> le L x 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 --> le L l 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. le L l 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. le L x 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 order_syntax
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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 order_syntax
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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 order_syntax
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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 order_syntax
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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 order_syntax
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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 order_syntax
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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 (unfold least_def, intro conjI)
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  show "Upper L A \<subseteq> carrier L" by simp
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next
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  from above L show "s \<in> Upper L A" by (simp add: Upper_def)
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next
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  from below show "ALL x : Upper L A. s \<sqsubseteq> x" by fast
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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 order_syntax
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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 order_syntax
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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 (unfold greatest_def, intro conjI)
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  show "Lower L A \<subseteq> carrier L" by simp
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next
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  from below L show "i \<in> Lower L A" by (simp add: Lower_def)
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next
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  from above show "ALL x : Lower L A. x \<sqsubseteq> i" by fast
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qed
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subsection {* Lattices *}
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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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    and inf_of_two_exists:
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    "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. greatest L s (Lower L {x, y})"
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lemma least_Upper_above:
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  includes order_syntax
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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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  by (unfold least_def) blast
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lemma greatest_Lower_above:
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  includes order_syntax
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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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  by (unfold greatest_def) blast
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subsubsection {* Supremum *}
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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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  ==> P (x \<squnion> y)"
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proof (unfold join_def sup_def)
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  assume L: "x \<in> carrier L" "y \<in> carrier L"
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    and P: "!!l. least L l (Upper L {x, y}) ==> P l"
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  with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
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  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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qed
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lemma (in lattice) join_closed [simp]:
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  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<squnion> y \<in> carrier L"
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  by (rule joinI) (rule least_carrier)
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lemma (in partial_order) sup_of_singletonI:      (* only reflexivity needed ? *)
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  "x \<in> carrier L ==> least L x (Upper L {x})"
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  by (rule least_UpperI) fast+
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lemma (in partial_order) sup_of_singleton [simp]:
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  includes order_syntax
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  shows "x \<in> carrier L ==> \<Squnion> {x} = x"
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  by (unfold sup_def) (blast intro: least_unique least_UpperI sup_of_singletonI)
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text {* Condition on A: supremum exists. *}
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lemma (in lattice) sup_insertI:
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  "[| !!s. least L s (Upper L (insert x A)) ==> P s;
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  least L a (Upper L A); x \<in> carrier L; A \<subseteq> carrier L |]
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  ==> P (\<Squnion> (insert x A))"
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proof (unfold sup_def)
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  assume L: "x \<in> carrier L" "A \<subseteq> carrier L"
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    and P: "!!l. least L l (Upper L (insert x A)) ==> P l"
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    and least_a: "least L a (Upper L A)"
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  from L least_a have La: "a \<in> carrier L" by simp
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  from L sup_of_two_exists least_a
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  obtain s where least_s: "least L s (Upper L {a, x})" by blast
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  show "P (THE l. least L l (Upper L (insert x A)))"
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  proof (rule theI2 [where a = s])
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    show "least L s (Upper L (insert x A))"
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    proof (rule least_UpperI)
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      fix z
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      assume xA: "z \<in> insert x A"
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      show "z \<sqsubseteq> s"
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      proof -
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	{
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	  assume "z = x" then have ?thesis
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	    by (simp add: least_Upper_above [OF least_s] L La)
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        }
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	moreover
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        {
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	  assume "z \<in> A"
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          with L least_s least_a have ?thesis
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	    by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
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        }
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      moreover note xA
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      ultimately show ?thesis by blast
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    qed
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  next
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    fix y
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    assume y: "y \<in> Upper L (insert x A)"
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    show "s \<sqsubseteq> y"
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    proof (rule least_le [OF least_s], rule Upper_memI)
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      fix z
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      assume z: "z \<in> {a, x}"
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      show "z \<sqsubseteq> y"
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      proof -
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	{
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          have y': "y \<in> Upper L A"
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	    apply (rule subsetD [where A = "Upper L (insert x A)"])
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	    apply (rule Upper_antimono) apply clarify apply assumption
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	    done
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	  assume "z = a"
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	  with y' least_a have ?thesis by (fast dest: least_le)
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        }
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	moreover
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	{
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           assume "z = x"
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           with y L have ?thesis by blast
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        }
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        moreover note z
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        ultimately show ?thesis by blast
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      qed
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    qed (rule Upper_closed [THEN subsetD])
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  next
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    from L show "insert x A \<subseteq> carrier L" by simp
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  next
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    from least_s show "s \<in> carrier L" by simp
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  qed
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next
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    fix l
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    assume least_l: "least L l (Upper L (insert x A))"
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    show "l = s"
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    proof (rule least_unique)
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      show "least L s (Upper L (insert x A))"
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      proof (rule least_UpperI)
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	fix z
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	assume xA: "z \<in> insert x A"
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	show "z \<sqsubseteq> s"
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      proof -
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	{
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	  assume "z = x" then have ?thesis
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	    by (simp add: least_Upper_above [OF least_s] L La)
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        }
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	moreover
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        {
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	  assume "z \<in> A"
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          with L least_s least_a have ?thesis
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	    by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
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        }
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	  moreover note xA
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	  ultimately show ?thesis by blast
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	qed
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      next
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	fix y
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	assume y: "y \<in> Upper L (insert x A)"
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	show "s \<sqsubseteq> y"
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	proof (rule least_le [OF least_s], rule Upper_memI)
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	  fix z
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	  assume z: "z \<in> {a, x}"
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	  show "z \<sqsubseteq> y"
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	  proof -
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	    {
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          have y': "y \<in> Upper L A"
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	    apply (rule subsetD [where A = "Upper L (insert x A)"])
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	    apply (rule Upper_antimono) apply clarify apply assumption
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	    done
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	  assume "z = a"
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	  with y' least_a have ?thesis by (fast dest: least_le)
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        }
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	moreover
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	{
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           assume "z = x"
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           with y L have ?thesis by blast
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            }
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            moreover note z
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            ultimately show ?thesis by blast
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	  qed
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	qed (rule Upper_closed [THEN subsetD])
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      next
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	from L show "insert x A \<subseteq> carrier L" by simp
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      next
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	from least_s show "s \<in> carrier L" by simp
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      qed
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    qed
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  qed
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qed
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lemma (in lattice) finite_sup_least:
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  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> least L (\<Squnion> A) (Upper L A)"
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proof (induct set: Finites)
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  case empty then show ?case by simp
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next
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  case (insert A x)
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  show ?case
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  proof (cases "A = {}")
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    case True
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    with insert show ?thesis by (simp add: sup_of_singletonI)
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   349
  next
ballarin@14551
   350
    case False
ballarin@14551
   351
    from insert show ?thesis
ballarin@14551
   352
    proof (rule_tac sup_insertI)
ballarin@14551
   353
      from False insert show "least L (\<Squnion> A) (Upper L A)" by simp
ballarin@14551
   354
    qed simp_all
ballarin@14551
   355
  qed
ballarin@14551
   356
qed
ballarin@14551
   357
ballarin@14551
   358
lemma (in lattice) finite_sup_insertI:
ballarin@14551
   359
  assumes P: "!!l. least L l (Upper L (insert x A)) ==> P l"
ballarin@14551
   360
    and xA: "finite A" "x \<in> carrier L" "A \<subseteq> carrier L"
ballarin@14551
   361
  shows "P (\<Squnion> (insert x A))"
ballarin@14551
   362
proof (cases "A = {}")
ballarin@14551
   363
  case True with P and xA show ?thesis
ballarin@14551
   364
    by (simp add: sup_of_singletonI)
ballarin@14551
   365
next
ballarin@14551
   366
  case False with P and xA show ?thesis
ballarin@14551
   367
    by (simp add: sup_insertI finite_sup_least)
ballarin@14551
   368
qed
ballarin@14551
   369
ballarin@14551
   370
lemma (in lattice) finite_sup_closed:
ballarin@14551
   371
  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Squnion> A \<in> carrier L"
ballarin@14551
   372
proof (induct set: Finites)
ballarin@14551
   373
  case empty then show ?case by simp
ballarin@14551
   374
next
ballarin@14551
   375
  case (insert A x) then show ?case
ballarin@14551
   376
    by (rule_tac finite_sup_insertI) (simp_all)
ballarin@14551
   377
qed
ballarin@14551
   378
ballarin@14551
   379
lemma (in lattice) join_left:
ballarin@14551
   380
  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> x \<squnion> y"
ballarin@14551
   381
  by (rule joinI [folded join_def]) (blast dest: least_mem )
ballarin@14551
   382
ballarin@14551
   383
lemma (in lattice) join_right:
ballarin@14551
   384
  "[| x \<in> carrier L; y \<in> carrier L |] ==> y \<sqsubseteq> x \<squnion> y"
ballarin@14551
   385
  by (rule joinI [folded join_def]) (blast dest: least_mem )
ballarin@14551
   386
ballarin@14551
   387
lemma (in lattice) sup_of_two_least:
ballarin@14551
   388
  "[| x \<in> carrier L; y \<in> carrier L |] ==> least L (\<Squnion> {x, y}) (Upper L {x, y})"
ballarin@14551
   389
proof (unfold sup_def)
ballarin@14551
   390
  assume L: "x \<in> carrier L" "y \<in> carrier L"
ballarin@14551
   391
  with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
ballarin@14551
   392
  with L show "least L (THE xa. least L xa (Upper L {x, y})) (Upper L {x, y})"
ballarin@14551
   393
  by (fast intro: theI2 least_unique)  (* blast fails *)
ballarin@14551
   394
qed
ballarin@14551
   395
ballarin@14551
   396
lemma (in lattice) join_le:
ballarin@14551
   397
  assumes sub: "x \<sqsubseteq> z" "y \<sqsubseteq> z"
ballarin@14551
   398
    and L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
ballarin@14551
   399
  shows "x \<squnion> y \<sqsubseteq> z"
ballarin@14551
   400
proof (rule joinI)
ballarin@14551
   401
  fix s
ballarin@14551
   402
  assume "least L s (Upper L {x, y})"
ballarin@14551
   403
  with sub L show "s \<sqsubseteq> z" by (fast elim: least_le intro: Upper_memI)
ballarin@14551
   404
qed
ballarin@14551
   405
  
ballarin@14551
   406
lemma (in lattice) join_assoc_lemma:
ballarin@14551
   407
  assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
ballarin@14551
   408
  shows "x \<squnion> (y \<squnion> z) = \<Squnion> {x, y, z}"
ballarin@14551
   409
proof (rule finite_sup_insertI)
wenzelm@14651
   410
  -- {* The textbook argument in Jacobson I, p 457 *}
ballarin@14551
   411
  fix s
ballarin@14551
   412
  assume sup: "least L s (Upper L {x, y, z})"
ballarin@14551
   413
  show "x \<squnion> (y \<squnion> z) = s"
ballarin@14551
   414
  proof (rule anti_sym)
ballarin@14551
   415
    from sup L show "x \<squnion> (y \<squnion> z) \<sqsubseteq> s"
ballarin@14551
   416
      by (fastsimp intro!: join_le elim: least_Upper_above)
ballarin@14551
   417
  next
ballarin@14551
   418
    from sup L show "s \<sqsubseteq> x \<squnion> (y \<squnion> z)"
ballarin@14551
   419
    by (erule_tac least_le)
ballarin@14551
   420
      (blast intro!: Upper_memI intro: trans join_left join_right join_closed)
ballarin@14551
   421
  qed (simp_all add: L least_carrier [OF sup])
ballarin@14551
   422
qed (simp_all add: L)
ballarin@14551
   423
ballarin@14551
   424
lemma join_comm:
ballarin@14551
   425
  includes order_syntax
ballarin@14551
   426
  shows "x \<squnion> y = y \<squnion> x"
ballarin@14551
   427
  by (unfold join_def) (simp add: insert_commute)
ballarin@14551
   428
ballarin@14551
   429
lemma (in lattice) join_assoc:
ballarin@14551
   430
  assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
ballarin@14551
   431
  shows "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
ballarin@14551
   432
proof -
ballarin@14551
   433
  have "(x \<squnion> y) \<squnion> z = z \<squnion> (x \<squnion> y)" by (simp only: join_comm)
ballarin@14551
   434
  also from L have "... = \<Squnion> {z, x, y}" by (simp add: join_assoc_lemma)
ballarin@14551
   435
  also from L have "... = \<Squnion> {x, y, z}" by (simp add: insert_commute)
ballarin@14551
   436
  also from L have "... = x \<squnion> (y \<squnion> z)" by (simp add: join_assoc_lemma)
ballarin@14551
   437
  finally show ?thesis .
ballarin@14551
   438
qed
ballarin@14551
   439
ballarin@14551
   440
subsubsection {* Infimum *}
ballarin@14551
   441
ballarin@14551
   442
lemma (in lattice) meetI:
ballarin@14551
   443
  "[| !!i. greatest L i (Lower L {x, y}) ==> P i;
ballarin@14551
   444
  x \<in> carrier L; y \<in> carrier L |]
ballarin@14551
   445
  ==> P (x \<sqinter> y)"
ballarin@14551
   446
proof (unfold meet_def inf_def)
ballarin@14551
   447
  assume L: "x \<in> carrier L" "y \<in> carrier L"
ballarin@14551
   448
    and P: "!!g. greatest L g (Lower L {x, y}) ==> P g"
ballarin@14551
   449
  with inf_of_two_exists obtain i where "greatest L i (Lower L {x, y})" by fast
ballarin@14551
   450
  with L show "P (THE g. greatest L g (Lower L {x, y}))"
ballarin@14551
   451
  by (fast intro: theI2 greatest_unique P)
ballarin@14551
   452
qed
ballarin@14551
   453
ballarin@14551
   454
lemma (in lattice) meet_closed [simp]:
ballarin@14551
   455
  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<in> carrier L"
ballarin@14551
   456
  by (rule meetI) (rule greatest_carrier)
ballarin@14551
   457
wenzelm@14651
   458
lemma (in partial_order) inf_of_singletonI:      (* only reflexivity needed ? *)
ballarin@14551
   459
  "x \<in> carrier L ==> greatest L x (Lower L {x})"
ballarin@14551
   460
  by (rule greatest_LowerI) fast+
ballarin@14551
   461
ballarin@14551
   462
lemma (in partial_order) inf_of_singleton [simp]:
ballarin@14551
   463
  includes order_syntax
ballarin@14551
   464
  shows "x \<in> carrier L ==> \<Sqinter> {x} = x"
ballarin@14551
   465
  by (unfold inf_def) (blast intro: greatest_unique greatest_LowerI inf_of_singletonI)
ballarin@14551
   466
ballarin@14551
   467
text {* Condition on A: infimum exists. *}
ballarin@14551
   468
ballarin@14551
   469
lemma (in lattice) inf_insertI:
ballarin@14551
   470
  "[| !!i. greatest L i (Lower L (insert x A)) ==> P i;
ballarin@14551
   471
  greatest L a (Lower L A); x \<in> carrier L; A \<subseteq> carrier L |]
ballarin@14551
   472
  ==> P (\<Sqinter> (insert x A))"
ballarin@14551
   473
proof (unfold inf_def)
ballarin@14551
   474
  assume L: "x \<in> carrier L" "A \<subseteq> carrier L"
ballarin@14551
   475
    and P: "!!g. greatest L g (Lower L (insert x A)) ==> P g"
ballarin@14551
   476
    and greatest_a: "greatest L a (Lower L A)"
ballarin@14551
   477
  from L greatest_a have La: "a \<in> carrier L" by simp
ballarin@14551
   478
  from L inf_of_two_exists greatest_a
ballarin@14551
   479
  obtain i where greatest_i: "greatest L i (Lower L {a, x})" by blast
ballarin@14551
   480
  show "P (THE g. greatest L g (Lower L (insert x A)))"
ballarin@14551
   481
  proof (rule theI2 [where a = i])
ballarin@14551
   482
    show "greatest L i (Lower L (insert x A))"
ballarin@14551
   483
    proof (rule greatest_LowerI)
ballarin@14551
   484
      fix z
ballarin@14551
   485
      assume xA: "z \<in> insert x A"
ballarin@14551
   486
      show "i \<sqsubseteq> z"
ballarin@14551
   487
      proof -
ballarin@14551
   488
	{
ballarin@14551
   489
	  assume "z = x" then have ?thesis
ballarin@14551
   490
	    by (simp add: greatest_Lower_above [OF greatest_i] L La)
ballarin@14551
   491
        }
ballarin@14551
   492
	moreover
ballarin@14551
   493
        {
ballarin@14551
   494
	  assume "z \<in> A"
ballarin@14551
   495
          with L greatest_i greatest_a have ?thesis
ballarin@14551
   496
	    by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
ballarin@14551
   497
        }
ballarin@14551
   498
      moreover note xA
ballarin@14551
   499
      ultimately show ?thesis by blast
ballarin@14551
   500
    qed
ballarin@14551
   501
  next
ballarin@14551
   502
    fix y
ballarin@14551
   503
    assume y: "y \<in> Lower L (insert x A)"
ballarin@14551
   504
    show "y \<sqsubseteq> i"
ballarin@14551
   505
    proof (rule greatest_le [OF greatest_i], rule Lower_memI)
ballarin@14551
   506
      fix z
ballarin@14551
   507
      assume z: "z \<in> {a, x}"
ballarin@14551
   508
      show "y \<sqsubseteq> z"
ballarin@14551
   509
      proof -
ballarin@14551
   510
	{
ballarin@14551
   511
          have y': "y \<in> Lower L A"
ballarin@14551
   512
	    apply (rule subsetD [where A = "Lower L (insert x A)"])
ballarin@14551
   513
	    apply (rule Lower_antimono) apply clarify apply assumption
ballarin@14551
   514
	    done
ballarin@14551
   515
	  assume "z = a"
ballarin@14551
   516
	  with y' greatest_a have ?thesis by (fast dest: greatest_le)
ballarin@14551
   517
        }
ballarin@14551
   518
	moreover
ballarin@14551
   519
	{
ballarin@14551
   520
           assume "z = x"
ballarin@14551
   521
           with y L have ?thesis by blast
ballarin@14551
   522
        }
ballarin@14551
   523
        moreover note z
ballarin@14551
   524
        ultimately show ?thesis by blast
ballarin@14551
   525
      qed
ballarin@14551
   526
    qed (rule Lower_closed [THEN subsetD])
ballarin@14551
   527
  next
ballarin@14551
   528
    from L show "insert x A \<subseteq> carrier L" by simp
ballarin@14551
   529
  next
ballarin@14551
   530
    from greatest_i show "i \<in> carrier L" by simp
ballarin@14551
   531
  qed
ballarin@14551
   532
next
ballarin@14551
   533
    fix g
ballarin@14551
   534
    assume greatest_g: "greatest L g (Lower L (insert x A))"
ballarin@14551
   535
    show "g = i"
ballarin@14551
   536
    proof (rule greatest_unique)
ballarin@14551
   537
      show "greatest L i (Lower L (insert x A))"
ballarin@14551
   538
      proof (rule greatest_LowerI)
ballarin@14551
   539
	fix z
ballarin@14551
   540
	assume xA: "z \<in> insert x A"
ballarin@14551
   541
	show "i \<sqsubseteq> z"
ballarin@14551
   542
      proof -
ballarin@14551
   543
	{
ballarin@14551
   544
	  assume "z = x" then have ?thesis
ballarin@14551
   545
	    by (simp add: greatest_Lower_above [OF greatest_i] L La)
ballarin@14551
   546
        }
ballarin@14551
   547
	moreover
ballarin@14551
   548
        {
ballarin@14551
   549
	  assume "z \<in> A"
ballarin@14551
   550
          with L greatest_i greatest_a have ?thesis
ballarin@14551
   551
	    by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
ballarin@14551
   552
        }
ballarin@14551
   553
	  moreover note xA
ballarin@14551
   554
	  ultimately show ?thesis by blast
ballarin@14551
   555
	qed
ballarin@14551
   556
      next
ballarin@14551
   557
	fix y
ballarin@14551
   558
	assume y: "y \<in> Lower L (insert x A)"
ballarin@14551
   559
	show "y \<sqsubseteq> i"
ballarin@14551
   560
	proof (rule greatest_le [OF greatest_i], rule Lower_memI)
ballarin@14551
   561
	  fix z
ballarin@14551
   562
	  assume z: "z \<in> {a, x}"
ballarin@14551
   563
	  show "y \<sqsubseteq> z"
ballarin@14551
   564
	  proof -
ballarin@14551
   565
	    {
ballarin@14551
   566
          have y': "y \<in> Lower L A"
ballarin@14551
   567
	    apply (rule subsetD [where A = "Lower L (insert x A)"])
ballarin@14551
   568
	    apply (rule Lower_antimono) apply clarify apply assumption
ballarin@14551
   569
	    done
ballarin@14551
   570
	  assume "z = a"
ballarin@14551
   571
	  with y' greatest_a have ?thesis by (fast dest: greatest_le)
ballarin@14551
   572
        }
ballarin@14551
   573
	moreover
ballarin@14551
   574
	{
ballarin@14551
   575
           assume "z = x"
ballarin@14551
   576
           with y L have ?thesis by blast
ballarin@14551
   577
            }
ballarin@14551
   578
            moreover note z
ballarin@14551
   579
            ultimately show ?thesis by blast
ballarin@14551
   580
	  qed
ballarin@14551
   581
	qed (rule Lower_closed [THEN subsetD])
ballarin@14551
   582
      next
ballarin@14551
   583
	from L show "insert x A \<subseteq> carrier L" by simp
ballarin@14551
   584
      next
ballarin@14551
   585
	from greatest_i show "i \<in> carrier L" by simp
ballarin@14551
   586
      qed
ballarin@14551
   587
    qed
ballarin@14551
   588
  qed
ballarin@14551
   589
qed
ballarin@14551
   590
ballarin@14551
   591
lemma (in lattice) finite_inf_greatest:
ballarin@14551
   592
  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> greatest L (\<Sqinter> A) (Lower L A)"
ballarin@14551
   593
proof (induct set: Finites)
ballarin@14551
   594
  case empty then show ?case by simp
ballarin@14551
   595
next
ballarin@14551
   596
  case (insert A x)
ballarin@14551
   597
  show ?case
ballarin@14551
   598
  proof (cases "A = {}")
ballarin@14551
   599
    case True
ballarin@14551
   600
    with insert show ?thesis by (simp add: inf_of_singletonI)
ballarin@14551
   601
  next
ballarin@14551
   602
    case False
ballarin@14551
   603
    from insert show ?thesis
ballarin@14551
   604
    proof (rule_tac inf_insertI)
ballarin@14551
   605
      from False insert show "greatest L (\<Sqinter> A) (Lower L A)" by simp
ballarin@14551
   606
    qed simp_all
ballarin@14551
   607
  qed
ballarin@14551
   608
qed
ballarin@14551
   609
ballarin@14551
   610
lemma (in lattice) finite_inf_insertI:
ballarin@14551
   611
  assumes P: "!!i. greatest L i (Lower L (insert x A)) ==> P i"
ballarin@14551
   612
    and xA: "finite A" "x \<in> carrier L" "A \<subseteq> carrier L"
ballarin@14551
   613
  shows "P (\<Sqinter> (insert x A))"
ballarin@14551
   614
proof (cases "A = {}")
ballarin@14551
   615
  case True with P and xA show ?thesis
ballarin@14551
   616
    by (simp add: inf_of_singletonI)
ballarin@14551
   617
next
ballarin@14551
   618
  case False with P and xA show ?thesis
ballarin@14551
   619
    by (simp add: inf_insertI finite_inf_greatest)
ballarin@14551
   620
qed
ballarin@14551
   621
ballarin@14551
   622
lemma (in lattice) finite_inf_closed:
ballarin@14551
   623
  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Sqinter> A \<in> carrier L"
ballarin@14551
   624
proof (induct set: Finites)
ballarin@14551
   625
  case empty then show ?case by simp
ballarin@14551
   626
next
ballarin@14551
   627
  case (insert A x) then show ?case
ballarin@14551
   628
    by (rule_tac finite_inf_insertI) (simp_all)
ballarin@14551
   629
qed
ballarin@14551
   630
ballarin@14551
   631
lemma (in lattice) meet_left:
ballarin@14551
   632
  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> x"
ballarin@14551
   633
  by (rule meetI [folded meet_def]) (blast dest: greatest_mem )
ballarin@14551
   634
ballarin@14551
   635
lemma (in lattice) meet_right:
ballarin@14551
   636
  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> y"
ballarin@14551
   637
  by (rule meetI [folded meet_def]) (blast dest: greatest_mem )
ballarin@14551
   638
ballarin@14551
   639
lemma (in lattice) inf_of_two_greatest:
ballarin@14551
   640
  "[| x \<in> carrier L; y \<in> carrier L |] ==>
ballarin@14551
   641
  greatest L (\<Sqinter> {x, y}) (Lower L {x, y})"
ballarin@14551
   642
proof (unfold inf_def)
ballarin@14551
   643
  assume L: "x \<in> carrier L" "y \<in> carrier L"
ballarin@14551
   644
  with inf_of_two_exists obtain s where "greatest L s (Lower L {x, y})" by fast
ballarin@14551
   645
  with L
ballarin@14551
   646
  show "greatest L (THE xa. greatest L xa (Lower L {x, y})) (Lower L {x, y})"
ballarin@14551
   647
  by (fast intro: theI2 greatest_unique)  (* blast fails *)
ballarin@14551
   648
qed
ballarin@14551
   649
ballarin@14551
   650
lemma (in lattice) meet_le:
ballarin@14551
   651
  assumes sub: "z \<sqsubseteq> x" "z \<sqsubseteq> y"
ballarin@14551
   652
    and L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
ballarin@14551
   653
  shows "z \<sqsubseteq> x \<sqinter> y"
ballarin@14551
   654
proof (rule meetI)
ballarin@14551
   655
  fix i
ballarin@14551
   656
  assume "greatest L i (Lower L {x, y})"
ballarin@14551
   657
  with sub L show "z \<sqsubseteq> i" by (fast elim: greatest_le intro: Lower_memI)
ballarin@14551
   658
qed
ballarin@14551
   659
  
ballarin@14551
   660
lemma (in lattice) meet_assoc_lemma:
ballarin@14551
   661
  assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
ballarin@14551
   662
  shows "x \<sqinter> (y \<sqinter> z) = \<Sqinter> {x, y, z}"
ballarin@14551
   663
proof (rule finite_inf_insertI)
ballarin@14551
   664
  txt {* The textbook argument in Jacobson I, p 457 *}
ballarin@14551
   665
  fix i
ballarin@14551
   666
  assume inf: "greatest L i (Lower L {x, y, z})"
ballarin@14551
   667
  show "x \<sqinter> (y \<sqinter> z) = i"
ballarin@14551
   668
  proof (rule anti_sym)
ballarin@14551
   669
    from inf L show "i \<sqsubseteq> x \<sqinter> (y \<sqinter> z)"
ballarin@14551
   670
      by (fastsimp intro!: meet_le elim: greatest_Lower_above)
ballarin@14551
   671
  next
ballarin@14551
   672
    from inf L show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> i"
ballarin@14551
   673
    by (erule_tac greatest_le)
ballarin@14551
   674
      (blast intro!: Lower_memI intro: trans meet_left meet_right meet_closed)
ballarin@14551
   675
  qed (simp_all add: L greatest_carrier [OF inf])
ballarin@14551
   676
qed (simp_all add: L)
ballarin@14551
   677
ballarin@14551
   678
lemma meet_comm:
ballarin@14551
   679
  includes order_syntax
ballarin@14551
   680
  shows "x \<sqinter> y = y \<sqinter> x"
ballarin@14551
   681
  by (unfold meet_def) (simp add: insert_commute)
ballarin@14551
   682
ballarin@14551
   683
lemma (in lattice) meet_assoc:
ballarin@14551
   684
  assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
ballarin@14551
   685
  shows "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
ballarin@14551
   686
proof -
ballarin@14551
   687
  have "(x \<sqinter> y) \<sqinter> z = z \<sqinter> (x \<sqinter> y)" by (simp only: meet_comm)
ballarin@14551
   688
  also from L have "... = \<Sqinter> {z, x, y}" by (simp add: meet_assoc_lemma)
ballarin@14551
   689
  also from L have "... = \<Sqinter> {x, y, z}" by (simp add: insert_commute)
ballarin@14551
   690
  also from L have "... = x \<sqinter> (y \<sqinter> z)" by (simp add: meet_assoc_lemma)
ballarin@14551
   691
  finally show ?thesis .
ballarin@14551
   692
qed
ballarin@14551
   693
ballarin@14551
   694
subsection {* Total Orders *}
ballarin@14551
   695
ballarin@14551
   696
locale total_order = lattice +
ballarin@14551
   697
  assumes total: "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
ballarin@14551
   698
ballarin@14551
   699
text {* Introduction rule: the usual definition of total order *}
ballarin@14551
   700
ballarin@14551
   701
lemma (in partial_order) total_orderI:
ballarin@14551
   702
  assumes total: "!!x y. [| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
ballarin@14551
   703
  shows "total_order L"
ballarin@14551
   704
proof (rule total_order.intro)
ballarin@14551
   705
  show "lattice_axioms L"
ballarin@14551
   706
  proof (rule lattice_axioms.intro)
ballarin@14551
   707
    fix x y
ballarin@14551
   708
    assume L: "x \<in> carrier L" "y \<in> carrier L"
ballarin@14551
   709
    show "EX s. least L s (Upper L {x, y})"
ballarin@14551
   710
    proof -
ballarin@14551
   711
      note total L
ballarin@14551
   712
      moreover
ballarin@14551
   713
      {
ballarin@14551
   714
	assume "x \<sqsubseteq> y"
ballarin@14551
   715
        with L have "least L y (Upper L {x, y})"
ballarin@14551
   716
	  by (rule_tac least_UpperI) auto
ballarin@14551
   717
      }
ballarin@14551
   718
      moreover
ballarin@14551
   719
      {
ballarin@14551
   720
	assume "y \<sqsubseteq> x"
ballarin@14551
   721
        with L have "least L x (Upper L {x, y})"
ballarin@14551
   722
	  by (rule_tac least_UpperI) auto
ballarin@14551
   723
      }
ballarin@14551
   724
      ultimately show ?thesis by blast
ballarin@14551
   725
    qed
ballarin@14551
   726
  next
ballarin@14551
   727
    fix x y
ballarin@14551
   728
    assume L: "x \<in> carrier L" "y \<in> carrier L"
ballarin@14551
   729
    show "EX i. greatest L i (Lower L {x, y})"
ballarin@14551
   730
    proof -
ballarin@14551
   731
      note total L
ballarin@14551
   732
      moreover
ballarin@14551
   733
      {
ballarin@14551
   734
	assume "y \<sqsubseteq> x"
ballarin@14551
   735
        with L have "greatest L y (Lower L {x, y})"
ballarin@14551
   736
	  by (rule_tac greatest_LowerI) auto
ballarin@14551
   737
      }
ballarin@14551
   738
      moreover
ballarin@14551
   739
      {
ballarin@14551
   740
	assume "x \<sqsubseteq> y"
ballarin@14551
   741
        with L have "greatest L x (Lower L {x, y})"
ballarin@14551
   742
	  by (rule_tac greatest_LowerI) auto
ballarin@14551
   743
      }
ballarin@14551
   744
      ultimately show ?thesis by blast
ballarin@14551
   745
    qed
ballarin@14551
   746
  qed
ballarin@14551
   747
qed (assumption | rule total_order_axioms.intro)+
ballarin@14551
   748
ballarin@14551
   749
subsection {* Complete lattices *}
ballarin@14551
   750
ballarin@14551
   751
locale complete_lattice = lattice +
ballarin@14551
   752
  assumes sup_exists:
ballarin@14551
   753
    "[| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)"
ballarin@14551
   754
    and inf_exists:
ballarin@14551
   755
    "[| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)"
ballarin@14551
   756
ballarin@14551
   757
text {* Introduction rule: the usual definition of complete lattice *}
ballarin@14551
   758
ballarin@14551
   759
lemma (in partial_order) complete_latticeI:
ballarin@14551
   760
  assumes sup_exists:
ballarin@14551
   761
    "!!A. [| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)"
ballarin@14551
   762
    and inf_exists:
ballarin@14551
   763
    "!!A. [| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)"
ballarin@14551
   764
  shows "complete_lattice L"
ballarin@14551
   765
proof (rule complete_lattice.intro)
ballarin@14551
   766
  show "lattice_axioms L"
ballarin@14551
   767
  by (rule lattice_axioms.intro) (blast intro: sup_exists inf_exists)+
ballarin@14551
   768
qed (assumption | rule complete_lattice_axioms.intro)+
ballarin@14551
   769
wenzelm@14651
   770
constdefs (structure L)
wenzelm@14651
   771
  top :: "_ => 'a" ("\<top>\<index>")
wenzelm@14651
   772
  "\<top> == sup L (carrier L)"
ballarin@14551
   773
wenzelm@14651
   774
  bottom :: "_ => 'a" ("\<bottom>\<index>")
wenzelm@14651
   775
  "\<bottom> == inf L (carrier L)"
ballarin@14551
   776
ballarin@14551
   777
ballarin@14551
   778
lemma (in complete_lattice) supI:
ballarin@14551
   779
  "[| !!l. least L l (Upper L A) ==> P l; A \<subseteq> carrier L |]
wenzelm@14651
   780
  ==> P (\<Squnion>A)"
ballarin@14551
   781
proof (unfold sup_def)
ballarin@14551
   782
  assume L: "A \<subseteq> carrier L"
ballarin@14551
   783
    and P: "!!l. least L l (Upper L A) ==> P l"
ballarin@14551
   784
  with sup_exists obtain s where "least L s (Upper L A)" by blast
ballarin@14551
   785
  with L show "P (THE l. least L l (Upper L A))"
ballarin@14551
   786
  by (fast intro: theI2 least_unique P)
ballarin@14551
   787
qed
ballarin@14551
   788
ballarin@14551
   789
lemma (in complete_lattice) sup_closed [simp]:
ballarin@14551
   790
  "A \<subseteq> carrier L ==> \<Squnion> A \<in> carrier L"
ballarin@14551
   791
  by (rule supI) simp_all
ballarin@14551
   792
ballarin@14551
   793
lemma (in complete_lattice) top_closed [simp, intro]:
ballarin@14551
   794
  "\<top> \<in> carrier L"
ballarin@14551
   795
  by (unfold top_def) simp
ballarin@14551
   796
ballarin@14551
   797
lemma (in complete_lattice) infI:
ballarin@14551
   798
  "[| !!i. greatest L i (Lower L A) ==> P i; A \<subseteq> carrier L |]
ballarin@14551
   799
  ==> P (\<Sqinter> A)"
ballarin@14551
   800
proof (unfold inf_def)
ballarin@14551
   801
  assume L: "A \<subseteq> carrier L"
ballarin@14551
   802
    and P: "!!l. greatest L l (Lower L A) ==> P l"
ballarin@14551
   803
  with inf_exists obtain s where "greatest L s (Lower L A)" by blast
ballarin@14551
   804
  with L show "P (THE l. greatest L l (Lower L A))"
ballarin@14551
   805
  by (fast intro: theI2 greatest_unique P)
ballarin@14551
   806
qed
ballarin@14551
   807
ballarin@14551
   808
lemma (in complete_lattice) inf_closed [simp]:
ballarin@14551
   809
  "A \<subseteq> carrier L ==> \<Sqinter> A \<in> carrier L"
ballarin@14551
   810
  by (rule infI) simp_all
ballarin@14551
   811
ballarin@14551
   812
lemma (in complete_lattice) bottom_closed [simp, intro]:
ballarin@14551
   813
  "\<bottom> \<in> carrier L"
ballarin@14551
   814
  by (unfold bottom_def) simp
ballarin@14551
   815
ballarin@14551
   816
text {* Jacobson: Theorem 8.1 *}
ballarin@14551
   817
ballarin@14551
   818
lemma Lower_empty [simp]:
ballarin@14551
   819
  "Lower L {} = carrier L"
ballarin@14551
   820
  by (unfold Lower_def) simp
ballarin@14551
   821
ballarin@14551
   822
lemma Upper_empty [simp]:
ballarin@14551
   823
  "Upper L {} = carrier L"
ballarin@14551
   824
  by (unfold Upper_def) simp
ballarin@14551
   825
ballarin@14551
   826
theorem (in partial_order) complete_lattice_criterion1:
ballarin@14551
   827
  assumes top_exists: "EX g. greatest L g (carrier L)"
ballarin@14551
   828
    and inf_exists:
ballarin@14551
   829
      "!!A. [| A \<subseteq> carrier L; A ~= {} |] ==> EX i. greatest L i (Lower L A)"
ballarin@14551
   830
  shows "complete_lattice L"
ballarin@14551
   831
proof (rule complete_latticeI)
ballarin@14551
   832
  from top_exists obtain top where top: "greatest L top (carrier L)" ..
ballarin@14551
   833
  fix A
ballarin@14551
   834
  assume L: "A \<subseteq> carrier L"
ballarin@14551
   835
  let ?B = "Upper L A"
ballarin@14551
   836
  from L top have "top \<in> ?B" by (fast intro!: Upper_memI intro: greatest_le)
ballarin@14551
   837
  then have B_non_empty: "?B ~= {}" by fast
ballarin@14551
   838
  have B_L: "?B \<subseteq> carrier L" by simp
ballarin@14551
   839
  from inf_exists [OF B_L B_non_empty]
ballarin@14551
   840
  obtain b where b_inf_B: "greatest L b (Lower L ?B)" ..
ballarin@14551
   841
  have "least L b (Upper L A)"
ballarin@14551
   842
apply (rule least_UpperI)
ballarin@14551
   843
   apply (rule greatest_le [where A = "Lower L ?B"]) 
ballarin@14551
   844
    apply (rule b_inf_B)
ballarin@14551
   845
   apply (rule Lower_memI)
ballarin@14551
   846
    apply (erule UpperD)
ballarin@14551
   847
     apply assumption
ballarin@14551
   848
    apply (rule L)
ballarin@14551
   849
   apply (fast intro: L [THEN subsetD])
ballarin@14551
   850
  apply (erule greatest_Lower_above [OF b_inf_B])
ballarin@14551
   851
  apply simp
ballarin@14551
   852
 apply (rule L)
ballarin@14551
   853
apply (rule greatest_carrier [OF b_inf_B]) (* rename rule: _closed *)
ballarin@14551
   854
done
ballarin@14551
   855
  then show "EX s. least L s (Upper L A)" ..
ballarin@14551
   856
next
ballarin@14551
   857
  fix A
ballarin@14551
   858
  assume L: "A \<subseteq> carrier L"
ballarin@14551
   859
  show "EX i. greatest L i (Lower L A)"
ballarin@14551
   860
  proof (cases "A = {}")
ballarin@14551
   861
    case True then show ?thesis
ballarin@14551
   862
      by (simp add: top_exists)
ballarin@14551
   863
  next
ballarin@14551
   864
    case False with L show ?thesis
ballarin@14551
   865
      by (rule inf_exists)
ballarin@14551
   866
  qed
ballarin@14551
   867
qed
ballarin@14551
   868
ballarin@14551
   869
(* TODO: prove dual version *)
ballarin@14551
   870
ballarin@14551
   871
subsection {* Examples *}
ballarin@14551
   872
ballarin@14551
   873
subsubsection {* Powerset of a set is a complete lattice *}
ballarin@14551
   874
ballarin@14551
   875
theorem powerset_is_complete_lattice:
ballarin@14551
   876
  "complete_lattice (| carrier = Pow A, le = op \<subseteq> |)"
ballarin@14551
   877
  (is "complete_lattice ?L")
ballarin@14551
   878
proof (rule partial_order.complete_latticeI)
ballarin@14551
   879
  show "partial_order ?L"
ballarin@14551
   880
    by (rule partial_order.intro) auto
ballarin@14551
   881
next
ballarin@14551
   882
  fix B
ballarin@14551
   883
  assume "B \<subseteq> carrier ?L"
ballarin@14551
   884
  then have "least ?L (\<Union> B) (Upper ?L B)"
ballarin@14551
   885
    by (fastsimp intro!: least_UpperI simp: Upper_def)
ballarin@14551
   886
  then show "EX s. least ?L s (Upper ?L B)" ..
ballarin@14551
   887
next
ballarin@14551
   888
  fix B
ballarin@14551
   889
  assume "B \<subseteq> carrier ?L"
ballarin@14551
   890
  then have "greatest ?L (\<Inter> B \<inter> A) (Lower ?L B)"
ballarin@14551
   891
    txt {* @{term "\<Inter> B"} is not the infimum of @{term B}:
ballarin@14551
   892
      @{term "\<Inter> {} = UNIV"} which is in general bigger than @{term "A"}! *}
ballarin@14551
   893
    by (fastsimp intro!: greatest_LowerI simp: Lower_def)
ballarin@14551
   894
  then show "EX i. greatest ?L i (Lower ?L B)" ..
ballarin@14551
   895
qed
ballarin@14551
   896
ballarin@14551
   897
subsubsection {* Lattice of subgroups of a group *}
ballarin@14551
   898
ballarin@14551
   899
theorem (in group) subgroups_partial_order:
ballarin@14551
   900
  "partial_order (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
ballarin@14551
   901
  by (rule partial_order.intro) simp_all
ballarin@14551
   902
ballarin@14551
   903
lemma (in group) subgroup_self:
ballarin@14551
   904
  "subgroup (carrier G) G"
ballarin@14551
   905
  by (rule subgroupI) auto
ballarin@14551
   906
ballarin@14551
   907
lemma (in group) subgroup_imp_group:
ballarin@14551
   908
  "subgroup H G ==> group (G(| carrier := H |))"
ballarin@14551
   909
  using subgroup.groupI [OF _ group.intro] .
ballarin@14551
   910
ballarin@14551
   911
lemma (in group) is_monoid [intro, simp]:
ballarin@14551
   912
  "monoid G"
ballarin@14551
   913
  by (rule monoid.intro)
ballarin@14551
   914
ballarin@14551
   915
lemma (in group) subgroup_inv_equality:
ballarin@14551
   916
  "[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x"
ballarin@14551
   917
apply (rule_tac inv_equality [THEN sym])
ballarin@14551
   918
  apply (rule group.l_inv [OF subgroup_imp_group, simplified])
ballarin@14551
   919
   apply assumption+
ballarin@14551
   920
 apply (rule subsetD [OF subgroup.subset])
ballarin@14551
   921
  apply assumption+
ballarin@14551
   922
apply (rule subsetD [OF subgroup.subset])
ballarin@14551
   923
 apply assumption
ballarin@14551
   924
apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified])
ballarin@14551
   925
  apply assumption+
ballarin@14551
   926
done
ballarin@14551
   927
ballarin@14551
   928
theorem (in group) subgroups_Inter:
ballarin@14551
   929
  assumes subgr: "(!!H. H \<in> A ==> subgroup H G)"
ballarin@14551
   930
    and not_empty: "A ~= {}"
ballarin@14551
   931
  shows "subgroup (\<Inter>A) G"
ballarin@14551
   932
proof (rule subgroupI)
ballarin@14551
   933
  from subgr [THEN subgroup.subset] and not_empty
ballarin@14551
   934
  show "\<Inter>A \<subseteq> carrier G" by blast
ballarin@14551
   935
next
ballarin@14551
   936
  from subgr [THEN subgroup.one_closed]
ballarin@14551
   937
  show "\<Inter>A ~= {}" by blast
ballarin@14551
   938
next
ballarin@14551
   939
  fix x assume "x \<in> \<Inter>A"
ballarin@14551
   940
  with subgr [THEN subgroup.m_inv_closed]
ballarin@14551
   941
  show "inv x \<in> \<Inter>A" by blast
ballarin@14551
   942
next
ballarin@14551
   943
  fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A"
ballarin@14551
   944
  with subgr [THEN subgroup.m_closed]
ballarin@14551
   945
  show "x \<otimes> y \<in> \<Inter>A" by blast
ballarin@14551
   946
qed
ballarin@14551
   947
ballarin@14551
   948
theorem (in group) subgroups_complete_lattice:
ballarin@14551
   949
  "complete_lattice (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
ballarin@14551
   950
    (is "complete_lattice ?L")
ballarin@14551
   951
proof (rule partial_order.complete_lattice_criterion1)
ballarin@14551
   952
  show "partial_order ?L" by (rule subgroups_partial_order)
ballarin@14551
   953
next
ballarin@14551
   954
  have "greatest ?L (carrier G) (carrier ?L)"
ballarin@14551
   955
    by (unfold greatest_def) (simp add: subgroup.subset subgroup_self)
ballarin@14551
   956
  then show "EX G. greatest ?L G (carrier ?L)" ..
ballarin@14551
   957
next
ballarin@14551
   958
  fix A
ballarin@14551
   959
  assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}"
ballarin@14551
   960
  then have Int_subgroup: "subgroup (\<Inter>A) G"
ballarin@14551
   961
    by (fastsimp intro: subgroups_Inter)
ballarin@14551
   962
  have "greatest ?L (\<Inter>A) (Lower ?L A)"
ballarin@14551
   963
    (is "greatest ?L ?Int _")
ballarin@14551
   964
  proof (rule greatest_LowerI)
ballarin@14551
   965
    fix H
ballarin@14551
   966
    assume H: "H \<in> A"
ballarin@14551
   967
    with L have subgroupH: "subgroup H G" by auto
ballarin@14551
   968
    from subgroupH have submagmaH: "submagma H G" by (rule subgroup.axioms)
ballarin@14551
   969
    from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H")
ballarin@14551
   970
      by (rule subgroup_imp_group)
ballarin@14551
   971
    from groupH have monoidH: "monoid ?H"
ballarin@14551
   972
      by (rule group.is_monoid)
ballarin@14551
   973
    from H have Int_subset: "?Int \<subseteq> H" by fastsimp
ballarin@14551
   974
    then show "le ?L ?Int H" by simp
ballarin@14551
   975
  next
ballarin@14551
   976
    fix H
ballarin@14551
   977
    assume H: "H \<in> Lower ?L A"
ballarin@14551
   978
    with L Int_subgroup show "le ?L H ?Int" by (fastsimp intro: Inter_greatest)
ballarin@14551
   979
  next
ballarin@14551
   980
    show "A \<subseteq> carrier ?L" by (rule L)
ballarin@14551
   981
  next
ballarin@14551
   982
    show "?Int \<in> carrier ?L" by simp (rule Int_subgroup)
ballarin@14551
   983
  qed
ballarin@14551
   984
  then show "EX I. greatest ?L I (Lower ?L A)" ..
ballarin@14551
   985
qed
ballarin@14551
   986
ballarin@14551
   987
end