src/HOL/Algebra/Lattice.thy
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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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theory Lattice imports Main begin
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section {* Orders and Lattices *}
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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 =
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  fixes L (structure)
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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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  lless :: "[_, '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>" 70)
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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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  fixes L (structure)
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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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  fixes L (structure)
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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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  fixes L (structure)
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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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  fixes L (structure)
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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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  fixes L (structure)
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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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  fixes L (structure)
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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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  fixes L (structure)
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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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  fixes L (structure)
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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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   166
    and L: "A \<subseteq> carrier L"  "i \<in> carrier L"
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ballarin
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   167
  shows "greatest L i (Lower L A)"
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   168
proof -
22063
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parents: 21896
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   169
  have "Lower L A \<subseteq> carrier L" by simp
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
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   170
  moreover from below L have "i \<in> Lower L A" by (simp add: Lower_def)
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ballarin
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   171
  moreover from above have "ALL x : Lower L A. x \<sqsubseteq> i" by fast
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   172
  ultimately show ?thesis by (simp add: greatest_def)
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parents:
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   173
qed
2cb6ff394bfb Various changes to HOL-Algebra;
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parents:
diff changeset
   174
14693
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   175
14551
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   176
subsection {* Lattices *}
2cb6ff394bfb Various changes to HOL-Algebra;
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   177
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
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   178
locale lattice = partial_order +
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   179
  assumes sup_of_two_exists:
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   180
    "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. least L s (Upper L {x, y})"
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   181
    and inf_of_two_exists:
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   182
    "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. greatest L s (Lower L {x, y})"
14551
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parents:
diff changeset
   183
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   184
lemma least_Upper_above:
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   185
  fixes L (structure)
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parents: 21896
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   186
  shows "[| least L s (Upper L A); x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> s"
14551
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ballarin
parents:
diff changeset
   187
  by (unfold least_def) blast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   188
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diff changeset
   189
lemma greatest_Lower_above:
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ballarin
parents: 21896
diff changeset
   190
  fixes L (structure)
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   191
  shows "[| greatest L i (Lower L A); x \<in> A; A \<subseteq> carrier L |] ==> i \<sqsubseteq> x"
14551
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ballarin
parents:
diff changeset
   192
  by (unfold greatest_def) blast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   193
14666
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   194
14551
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   195
subsubsection {* Supremum *}
2cb6ff394bfb Various changes to HOL-Algebra;
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   196
2cb6ff394bfb Various changes to HOL-Algebra;
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diff changeset
   197
lemma (in lattice) joinI:
22063
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   198
  "[| !!l. least L l (Upper L {x, y}) ==> P l; x \<in> carrier L; y \<in> carrier L |]
14551
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   199
  ==> P (x \<squnion> y)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   200
proof (unfold join_def sup_def)
22063
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   201
  assume L: "x \<in> carrier L"  "y \<in> carrier L"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   202
    and P: "!!l. least L l (Upper L {x, y}) ==> P l"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   203
  with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   204
  with L show "P (THE l. least L l (Upper L {x, y}))"
14693
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   205
    by (fast intro: theI2 least_unique P)
14551
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parents:
diff changeset
   206
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   207
2cb6ff394bfb Various changes to HOL-Algebra;
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parents:
diff changeset
   208
lemma (in lattice) join_closed [simp]:
22063
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   209
  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<squnion> y \<in> carrier L"
717425609192 Reverted to structure representation with records.
ballarin
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   210
  by (rule joinI) (rule least_carrier)
14551
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ballarin
parents:
diff changeset
   211
22063
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ballarin
parents: 21896
diff changeset
   212
lemma (in partial_order) sup_of_singletonI:      (* only reflexivity needed ? *)
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   213
  "x \<in> carrier L ==> least L x (Upper L {x})"
14551
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ballarin
parents:
diff changeset
   214
  by (rule least_UpperI) fast+
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   215
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   216
lemma (in partial_order) sup_of_singleton [simp]:
22063
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ballarin
parents: 21896
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   217
  "x \<in> carrier L ==> \<Squnion>{x} = x"
14551
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ballarin
parents:
diff changeset
   218
  by (unfold sup_def) (blast intro: least_unique least_UpperI sup_of_singletonI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   219
14666
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   220
65f8680c3f16 improved notation;
wenzelm
parents: 14651
diff changeset
   221
text {* Condition on @{text A}: supremum exists. *}
14551
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ballarin
parents:
diff changeset
   222
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   223
lemma (in lattice) sup_insertI:
22063
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ballarin
parents: 21896
diff changeset
   224
  "[| !!s. least L s (Upper L (insert x A)) ==> P s;
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   225
  least L a (Upper L A); x \<in> carrier L; A \<subseteq> carrier L |]
14693
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wenzelm
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diff changeset
   226
  ==> P (\<Squnion>(insert x A))"
14551
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ballarin
parents:
diff changeset
   227
proof (unfold sup_def)
22063
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ballarin
parents: 21896
diff changeset
   228
  assume L: "x \<in> carrier L"  "A \<subseteq> carrier L"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   229
    and P: "!!l. least L l (Upper L (insert x A)) ==> P l"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   230
    and least_a: "least L a (Upper L A)"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   231
  from L least_a have La: "a \<in> carrier L" by simp
14551
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ballarin
parents:
diff changeset
   232
  from L sup_of_two_exists least_a
22063
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ballarin
parents: 21896
diff changeset
   233
  obtain s where least_s: "least L s (Upper L {a, x})" by blast
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   234
  show "P (THE l. least L l (Upper L (insert x A)))"
14693
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wenzelm
parents: 14666
diff changeset
   235
  proof (rule theI2)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   236
    show "least L s (Upper L (insert x A))"
14551
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ballarin
parents:
diff changeset
   237
    proof (rule least_UpperI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   238
      fix z
14693
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wenzelm
parents: 14666
diff changeset
   239
      assume "z \<in> insert x A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   240
      then show "z \<sqsubseteq> s"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   241
      proof
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   242
        assume "z = x" then show ?thesis
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   243
          by (simp add: least_Upper_above [OF least_s] L La)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   244
      next
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   245
        assume "z \<in> A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   246
        with L least_s least_a show ?thesis
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   247
          by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   248
      qed
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   249
    next
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   250
      fix y
22063
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ballarin
parents: 21896
diff changeset
   251
      assume y: "y \<in> Upper L (insert x A)"
14693
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wenzelm
parents: 14666
diff changeset
   252
      show "s \<sqsubseteq> y"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   253
      proof (rule least_le [OF least_s], rule Upper_memI)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   254
	fix z
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   255
	assume z: "z \<in> {a, x}"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   256
	then show "z \<sqsubseteq> y"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   257
	proof
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   258
          have y': "y \<in> Upper L A"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   259
            apply (rule subsetD [where A = "Upper L (insert x A)"])
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   260
            apply (rule Upper_antimono) apply clarify apply assumption
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   261
            done
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   262
          assume "z = a"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   263
          with y' least_a show ?thesis by (fast dest: least_le)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   264
	next
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   265
	  assume "z \<in> {x}"  (* FIXME "z = x"; declare specific elim rule for "insert x {}" (!?) *)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   266
          with y L show ?thesis by blast
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   267
	qed
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22265
diff changeset
   268
      qed (rule Upper_closed [THEN subsetD, OF y])
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   269
    next
22063
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ballarin
parents: 21896
diff changeset
   270
      from L show "insert x A \<subseteq> carrier L" by simp
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   271
      from least_s show "s \<in> carrier L" by simp
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   272
    qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   273
  next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   274
    fix l
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   275
    assume least_l: "least L l (Upper L (insert x A))"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   276
    show "l = s"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   277
    proof (rule least_unique)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   278
      show "least L s (Upper L (insert x A))"
14551
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ballarin
parents:
diff changeset
   279
      proof (rule least_UpperI)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   280
        fix z
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   281
        assume "z \<in> insert x A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   282
        then show "z \<sqsubseteq> s"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   283
	proof
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   284
          assume "z = x" then show ?thesis
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   285
            by (simp add: least_Upper_above [OF least_s] L La)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   286
	next
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   287
          assume "z \<in> A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   288
          with L least_s least_a show ?thesis
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   289
            by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   290
	qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   291
      next
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   292
        fix y
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   293
        assume y: "y \<in> Upper L (insert x A)"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   294
        show "s \<sqsubseteq> y"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   295
        proof (rule least_le [OF least_s], rule Upper_memI)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   296
          fix z
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   297
          assume z: "z \<in> {a, x}"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   298
          then show "z \<sqsubseteq> y"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   299
          proof
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   300
            have y': "y \<in> Upper L A"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   301
	      apply (rule subsetD [where A = "Upper L (insert x A)"])
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   302
	      apply (rule Upper_antimono) apply clarify apply assumption
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   303
	      done
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   304
            assume "z = a"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   305
            with y' least_a show ?thesis by (fast dest: least_le)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   306
	  next
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   307
            assume "z \<in> {x}"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   308
            with y L show ?thesis by blast
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   309
          qed
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22265
diff changeset
   310
        qed (rule Upper_closed [THEN subsetD, OF y])
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   311
      next
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   312
        from L show "insert x A \<subseteq> carrier L" by simp
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   313
        from least_s show "s \<in> carrier L" by simp
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   314
      qed
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22265
diff changeset
   315
    qed (rule least_l)
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22265
diff changeset
   316
  qed (rule P)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   317
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   318
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   319
lemma (in lattice) finite_sup_least:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   320
  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> least L (\<Squnion>A) (Upper L A)"
22265
3c5c6bdf61de Adapted to changes in Finite_Set theory.
berghofe
parents: 22063
diff changeset
   321
proof (induct set: finite)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   322
  case empty
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   323
  then show ?case by simp
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   324
next
15328
35951e6a7855 mod because of change in finite set induction
nipkow
parents: 14751
diff changeset
   325
  case (insert x A)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   326
  show ?case
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   327
  proof (cases "A = {}")
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   328
    case True
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   329
    with insert show ?thesis by (simp add: sup_of_singletonI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   330
  next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   331
    case False
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   332
    with insert have "least L (\<Squnion>A) (Upper L A)" by simp
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   333
    with _ show ?thesis
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   334
      by (rule sup_insertI) (simp_all add: insert [simplified])
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   335
  qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   336
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   337
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   338
lemma (in lattice) finite_sup_insertI:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   339
  assumes P: "!!l. least L l (Upper L (insert x A)) ==> P l"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   340
    and xA: "finite A"  "x \<in> carrier L"  "A \<subseteq> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   341
  shows "P (\<Squnion> (insert x A))"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   342
proof (cases "A = {}")
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   343
  case True with P and xA show ?thesis
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   344
    by (simp add: sup_of_singletonI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   345
next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   346
  case False with P and xA show ?thesis
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   347
    by (simp add: sup_insertI finite_sup_least)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   348
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   349
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   350
lemma (in lattice) finite_sup_closed:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   351
  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Squnion>A \<in> carrier L"
22265
3c5c6bdf61de Adapted to changes in Finite_Set theory.
berghofe
parents: 22063
diff changeset
   352
proof (induct set: finite)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   353
  case empty then show ?case by simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   354
next
15328
35951e6a7855 mod because of change in finite set induction
nipkow
parents: 14751
diff changeset
   355
  case insert then show ?case
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   356
    by - (rule finite_sup_insertI, simp_all)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   357
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   358
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   359
lemma (in lattice) join_left:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   360
  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> x \<squnion> y"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   361
  by (rule joinI [folded join_def]) (blast dest: least_mem)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   362
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   363
lemma (in lattice) join_right:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   364
  "[| x \<in> carrier L; y \<in> carrier L |] ==> y \<sqsubseteq> x \<squnion> y"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   365
  by (rule joinI [folded join_def]) (blast dest: least_mem)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   366
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   367
lemma (in lattice) sup_of_two_least:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   368
  "[| 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
   369
proof (unfold sup_def)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   370
  assume L: "x \<in> carrier L"  "y \<in> carrier L"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   371
  with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   372
  with L show "least L (THE xa. least L xa (Upper L {x, y})) (Upper L {x, y})"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   373
  by (fast intro: theI2 least_unique)  (* blast fails *)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   374
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   375
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   376
lemma (in lattice) join_le:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   377
  assumes sub: "x \<sqsubseteq> z"  "y \<sqsubseteq> z"
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22265
diff changeset
   378
    and x: "x \<in> carrier L" and y: "y \<in> carrier L" and z: "z \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   379
  shows "x \<squnion> y \<sqsubseteq> z"
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22265
diff changeset
   380
proof (rule joinI [OF _ x y])
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   381
  fix s
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   382
  assume "least L s (Upper L {x, y})"
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22265
diff changeset
   383
  with sub z show "s \<sqsubseteq> z" by (fast elim: least_le intro: Upper_memI)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   384
qed
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   385
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   386
lemma (in lattice) join_assoc_lemma:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   387
  assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   388
  shows "x \<squnion> (y \<squnion> z) = \<Squnion>{x, y, z}"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   389
proof (rule finite_sup_insertI)
14651
02b8f3bcf7fe improved notation;
wenzelm
parents: 14577
diff changeset
   390
  -- {* The textbook argument in Jacobson I, p 457 *}
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   391
  fix s
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   392
  assume sup: "least L s (Upper L {x, y, z})"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   393
  show "x \<squnion> (y \<squnion> z) = s"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   394
  proof (rule anti_sym)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   395
    from sup L show "x \<squnion> (y \<squnion> z) \<sqsubseteq> s"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   396
      by (fastsimp intro!: join_le elim: least_Upper_above)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   397
  next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   398
    from sup L show "s \<sqsubseteq> x \<squnion> (y \<squnion> z)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   399
    by (erule_tac least_le)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   400
      (blast intro!: Upper_memI intro: trans join_left join_right join_closed)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   401
  qed (simp_all add: L least_carrier [OF sup])
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   402
qed (simp_all add: L)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   403
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   404
lemma join_comm:
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   405
  fixes L (structure)
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   406
  shows "x \<squnion> y = y \<squnion> x"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   407
  by (unfold join_def) (simp add: insert_commute)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   408
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   409
lemma (in lattice) join_assoc:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   410
  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
   411
  shows "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   412
proof -
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   413
  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
   414
  also from L have "... = \<Squnion>{z, x, y}" by (simp add: join_assoc_lemma)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   415
  also from L have "... = \<Squnion>{x, y, z}" by (simp add: insert_commute)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   416
  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
   417
  finally show ?thesis .
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   418
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   419
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   420
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   421
subsubsection {* Infimum *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   422
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   423
lemma (in lattice) meetI:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   424
  "[| !!i. greatest L i (Lower L {x, y}) ==> P i;
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   425
  x \<in> carrier L; y \<in> carrier L |]
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   426
  ==> P (x \<sqinter> y)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   427
proof (unfold meet_def inf_def)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   428
  assume L: "x \<in> carrier L"  "y \<in> carrier L"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   429
    and P: "!!g. greatest L g (Lower L {x, y}) ==> P g"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   430
  with inf_of_two_exists obtain i where "greatest L i (Lower L {x, y})" by fast
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   431
  with L show "P (THE g. greatest L g (Lower L {x, y}))"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   432
  by (fast intro: theI2 greatest_unique P)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   433
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   434
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   435
lemma (in lattice) meet_closed [simp]:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   436
  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<in> carrier L"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   437
  by (rule meetI) (rule greatest_carrier)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   438
14651
02b8f3bcf7fe improved notation;
wenzelm
parents: 14577
diff changeset
   439
lemma (in partial_order) inf_of_singletonI:      (* only reflexivity needed ? *)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   440
  "x \<in> carrier L ==> greatest L x (Lower L {x})"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   441
  by (rule greatest_LowerI) fast+
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   442
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   443
lemma (in partial_order) inf_of_singleton [simp]:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   444
  "x \<in> carrier L ==> \<Sqinter> {x} = x"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   445
  by (unfold inf_def) (blast intro: greatest_unique greatest_LowerI inf_of_singletonI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   446
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   447
text {* Condition on A: infimum exists. *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   448
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   449
lemma (in lattice) inf_insertI:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   450
  "[| !!i. greatest L i (Lower L (insert x A)) ==> P i;
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   451
  greatest L a (Lower L A); x \<in> carrier L; A \<subseteq> carrier L |]
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   452
  ==> P (\<Sqinter>(insert x A))"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   453
proof (unfold inf_def)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   454
  assume L: "x \<in> carrier L"  "A \<subseteq> carrier L"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   455
    and P: "!!g. greatest L g (Lower L (insert x A)) ==> P g"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   456
    and greatest_a: "greatest L a (Lower L A)"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   457
  from L greatest_a have La: "a \<in> carrier L" by simp
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   458
  from L inf_of_two_exists greatest_a
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   459
  obtain i where greatest_i: "greatest L i (Lower L {a, x})" by blast
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   460
  show "P (THE g. greatest L g (Lower L (insert x A)))"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   461
  proof (rule theI2)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   462
    show "greatest L i (Lower L (insert x A))"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   463
    proof (rule greatest_LowerI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   464
      fix z
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   465
      assume "z \<in> insert x A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   466
      then show "i \<sqsubseteq> z"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   467
      proof
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   468
        assume "z = x" then show ?thesis
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   469
          by (simp add: greatest_Lower_above [OF greatest_i] L La)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   470
      next
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   471
        assume "z \<in> A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   472
        with L greatest_i greatest_a show ?thesis
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   473
          by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   474
      qed
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   475
    next
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   476
      fix y
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   477
      assume y: "y \<in> Lower L (insert x A)"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   478
      show "y \<sqsubseteq> i"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   479
      proof (rule greatest_le [OF greatest_i], rule Lower_memI)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   480
	fix z
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   481
	assume z: "z \<in> {a, x}"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   482
	then show "y \<sqsubseteq> z"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   483
	proof
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   484
          have y': "y \<in> Lower L A"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   485
            apply (rule subsetD [where A = "Lower L (insert x A)"])
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   486
            apply (rule Lower_antimono) apply clarify apply assumption
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   487
            done
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   488
          assume "z = a"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   489
          with y' greatest_a show ?thesis by (fast dest: greatest_le)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   490
	next
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   491
          assume "z \<in> {x}"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   492
          with y L show ?thesis by blast
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   493
	qed
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22265
diff changeset
   494
      qed (rule Lower_closed [THEN subsetD, OF y])
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   495
    next
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   496
      from L show "insert x A \<subseteq> carrier L" by simp
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   497
      from greatest_i show "i \<in> carrier L" by simp
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   498
    qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   499
  next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   500
    fix g
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   501
    assume greatest_g: "greatest L g (Lower L (insert x A))"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   502
    show "g = i"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   503
    proof (rule greatest_unique)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   504
      show "greatest L i (Lower L (insert x A))"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   505
      proof (rule greatest_LowerI)
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   506
        fix z
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   507
        assume "z \<in> insert x A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   508
        then show "i \<sqsubseteq> z"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   509
	proof
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   510
          assume "z = x" then show ?thesis
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   511
            by (simp add: greatest_Lower_above [OF greatest_i] L La)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   512
	next
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   513
          assume "z \<in> A"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   514
          with L greatest_i greatest_a show ?thesis
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   515
            by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   516
        qed
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   517
      next
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   518
        fix y
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   519
        assume y: "y \<in> Lower L (insert x A)"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   520
        show "y \<sqsubseteq> i"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   521
        proof (rule greatest_le [OF greatest_i], rule Lower_memI)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   522
          fix z
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   523
          assume z: "z \<in> {a, x}"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   524
          then show "y \<sqsubseteq> z"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   525
          proof
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   526
            have y': "y \<in> Lower L A"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   527
	      apply (rule subsetD [where A = "Lower L (insert x A)"])
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   528
	      apply (rule Lower_antimono) apply clarify apply assumption
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   529
	      done
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   530
            assume "z = a"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   531
            with y' greatest_a show ?thesis by (fast dest: greatest_le)
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   532
	  next
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   533
            assume "z \<in> {x}"
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   534
            with y L show ?thesis by blast
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   535
	  qed
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22265
diff changeset
   536
        qed (rule Lower_closed [THEN subsetD, OF y])
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   537
      next
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   538
        from L show "insert x A \<subseteq> carrier L" by simp
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   539
        from greatest_i show "i \<in> carrier L" by simp
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   540
      qed
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22265
diff changeset
   541
    qed (rule greatest_g)
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22265
diff changeset
   542
  qed (rule P)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   543
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   544
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   545
lemma (in lattice) finite_inf_greatest:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   546
  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> greatest L (\<Sqinter>A) (Lower L A)"
22265
3c5c6bdf61de Adapted to changes in Finite_Set theory.
berghofe
parents: 22063
diff changeset
   547
proof (induct set: finite)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   548
  case empty then show ?case by simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   549
next
15328
35951e6a7855 mod because of change in finite set induction
nipkow
parents: 14751
diff changeset
   550
  case (insert x A)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   551
  show ?case
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   552
  proof (cases "A = {}")
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   553
    case True
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   554
    with insert show ?thesis by (simp add: inf_of_singletonI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   555
  next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   556
    case False
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   557
    from insert show ?thesis
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   558
    proof (rule_tac inf_insertI)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   559
      from False insert show "greatest L (\<Sqinter>A) (Lower L A)" by simp
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   560
    qed simp_all
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   561
  qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   562
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   563
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   564
lemma (in lattice) finite_inf_insertI:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   565
  assumes P: "!!i. greatest L i (Lower L (insert x A)) ==> P i"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   566
    and xA: "finite A"  "x \<in> carrier L"  "A \<subseteq> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   567
  shows "P (\<Sqinter> (insert x A))"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   568
proof (cases "A = {}")
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   569
  case True with P and xA show ?thesis
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   570
    by (simp add: inf_of_singletonI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   571
next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   572
  case False with P and xA show ?thesis
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   573
    by (simp add: inf_insertI finite_inf_greatest)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   574
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   575
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   576
lemma (in lattice) finite_inf_closed:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   577
  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Sqinter>A \<in> carrier L"
22265
3c5c6bdf61de Adapted to changes in Finite_Set theory.
berghofe
parents: 22063
diff changeset
   578
proof (induct set: finite)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   579
  case empty then show ?case by simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   580
next
15328
35951e6a7855 mod because of change in finite set induction
nipkow
parents: 14751
diff changeset
   581
  case insert then show ?case
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   582
    by (rule_tac finite_inf_insertI) (simp_all)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   583
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   584
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   585
lemma (in lattice) meet_left:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   586
  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> x"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   587
  by (rule meetI [folded meet_def]) (blast dest: greatest_mem)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   588
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   589
lemma (in lattice) meet_right:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   590
  "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> y"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   591
  by (rule meetI [folded meet_def]) (blast dest: greatest_mem)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   592
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   593
lemma (in lattice) inf_of_two_greatest:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   594
  "[| x \<in> carrier L; y \<in> carrier L |] ==>
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   595
  greatest L (\<Sqinter> {x, y}) (Lower L {x, y})"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   596
proof (unfold inf_def)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   597
  assume L: "x \<in> carrier L"  "y \<in> carrier L"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   598
  with inf_of_two_exists obtain s where "greatest L s (Lower L {x, y})" by fast
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   599
  with L
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   600
  show "greatest L (THE xa. greatest L xa (Lower L {x, y})) (Lower L {x, y})"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   601
  by (fast intro: theI2 greatest_unique)  (* blast fails *)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   602
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   603
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   604
lemma (in lattice) meet_le:
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   605
  assumes sub: "z \<sqsubseteq> x"  "z \<sqsubseteq> y"
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22265
diff changeset
   606
    and x: "x \<in> carrier L" and y: "y \<in> carrier L" and z: "z \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   607
  shows "z \<sqsubseteq> x \<sqinter> y"
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22265
diff changeset
   608
proof (rule meetI [OF _ x y])
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   609
  fix i
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   610
  assume "greatest L i (Lower L {x, y})"
23350
50c5b0912a0c tuned proofs: avoid implicit prems;
wenzelm
parents: 22265
diff changeset
   611
  with sub z show "z \<sqsubseteq> i" by (fast elim: greatest_le intro: Lower_memI)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   612
qed
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   613
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   614
lemma (in lattice) meet_assoc_lemma:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   615
  assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   616
  shows "x \<sqinter> (y \<sqinter> z) = \<Sqinter>{x, y, z}"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   617
proof (rule finite_inf_insertI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   618
  txt {* The textbook argument in Jacobson I, p 457 *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   619
  fix i
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   620
  assume inf: "greatest L i (Lower L {x, y, z})"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   621
  show "x \<sqinter> (y \<sqinter> z) = i"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   622
  proof (rule anti_sym)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   623
    from inf L show "i \<sqsubseteq> x \<sqinter> (y \<sqinter> z)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   624
      by (fastsimp intro!: meet_le elim: greatest_Lower_above)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   625
  next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   626
    from inf L show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> i"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   627
    by (erule_tac greatest_le)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   628
      (blast intro!: Lower_memI intro: trans meet_left meet_right meet_closed)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   629
  qed (simp_all add: L greatest_carrier [OF inf])
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   630
qed (simp_all add: L)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   631
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   632
lemma meet_comm:
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   633
  fixes L (structure)
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   634
  shows "x \<sqinter> y = y \<sqinter> x"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   635
  by (unfold meet_def) (simp add: insert_commute)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   636
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   637
lemma (in lattice) meet_assoc:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   638
  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
   639
  shows "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   640
proof -
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   641
  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
   642
  also from L have "... = \<Sqinter> {z, x, y}" by (simp add: meet_assoc_lemma)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   643
  also from L have "... = \<Sqinter> {x, y, z}" by (simp add: insert_commute)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   644
  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
   645
  finally show ?thesis .
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   646
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   647
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   648
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   649
subsection {* Total Orders *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   650
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   651
locale total_order = lattice +
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   652
  assumes total: "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   653
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   654
text {* Introduction rule: the usual definition of total order *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   655
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   656
lemma (in partial_order) total_orderI:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   657
  assumes total: "!!x y. [| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   658
  shows "total_order L"
19984
29bb4659f80a Method intro_locales replaced by intro_locales and unfold_locales.
ballarin
parents: 19931
diff changeset
   659
proof intro_locales
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   660
  show "lattice_axioms L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   661
  proof (rule lattice_axioms.intro)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   662
    fix x y
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   663
    assume L: "x \<in> carrier L"  "y \<in> carrier L"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   664
    show "EX s. least L s (Upper L {x, y})"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   665
    proof -
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   666
      note total L
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   667
      moreover
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   668
      {
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   669
        assume "x \<sqsubseteq> y"
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   670
        with L have "least L y (Upper L {x, y})"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   671
          by (rule_tac least_UpperI) auto
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   672
      }
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   673
      moreover
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   674
      {
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   675
        assume "y \<sqsubseteq> x"
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   676
        with L have "least L x (Upper L {x, y})"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   677
          by (rule_tac least_UpperI) auto
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   678
      }
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   679
      ultimately show ?thesis by blast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   680
    qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   681
  next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   682
    fix x y
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   683
    assume L: "x \<in> carrier L"  "y \<in> carrier L"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   684
    show "EX i. greatest L i (Lower L {x, y})"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   685
    proof -
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   686
      note total L
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   687
      moreover
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   688
      {
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   689
        assume "y \<sqsubseteq> x"
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   690
        with L have "greatest L y (Lower L {x, y})"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   691
          by (rule_tac greatest_LowerI) auto
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   692
      }
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   693
      moreover
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   694
      {
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   695
        assume "x \<sqsubseteq> y"
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   696
        with L have "greatest L x (Lower L {x, y})"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   697
          by (rule_tac greatest_LowerI) auto
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   698
      }
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   699
      ultimately show ?thesis by blast
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   700
    qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   701
  qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   702
qed (assumption | rule total_order_axioms.intro)+
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   703
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   704
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   705
subsection {* Complete lattices *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   706
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   707
locale complete_lattice = lattice +
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   708
  assumes sup_exists:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   709
    "[| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   710
    and inf_exists:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   711
    "[| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)"
21041
60e418260b4d Order and lattice structures no longer based on records.
ballarin
parents: 20318
diff changeset
   712
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   713
text {* Introduction rule: the usual definition of complete lattice *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   714
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   715
lemma (in partial_order) complete_latticeI:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   716
  assumes sup_exists:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   717
    "!!A. [| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   718
    and inf_exists:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   719
    "!!A. [| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   720
  shows "complete_lattice L"
19984
29bb4659f80a Method intro_locales replaced by intro_locales and unfold_locales.
ballarin
parents: 19931
diff changeset
   721
proof intro_locales
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   722
  show "lattice_axioms L"
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   723
    by (rule lattice_axioms.intro) (blast intro: sup_exists inf_exists)+
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   724
qed (assumption | rule complete_lattice_axioms.intro)+
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   725
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   726
constdefs (structure L)
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   727
  top :: "_ => 'a" ("\<top>\<index>")
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   728
  "\<top> == sup L (carrier L)"
21041
60e418260b4d Order and lattice structures no longer based on records.
ballarin
parents: 20318
diff changeset
   729
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   730
  bottom :: "_ => 'a" ("\<bottom>\<index>")
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   731
  "\<bottom> == inf L (carrier L)"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   732
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   733
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   734
lemma (in complete_lattice) supI:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   735
  "[| !!l. least L l (Upper L A) ==> P l; A \<subseteq> carrier L |]
14651
02b8f3bcf7fe improved notation;
wenzelm
parents: 14577
diff changeset
   736
  ==> P (\<Squnion>A)"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   737
proof (unfold sup_def)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   738
  assume L: "A \<subseteq> carrier L"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   739
    and P: "!!l. least L l (Upper L A) ==> P l"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   740
  with sup_exists obtain s where "least L s (Upper L A)" by blast
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   741
  with L show "P (THE l. least L l (Upper L A))"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   742
  by (fast intro: theI2 least_unique P)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   743
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   744
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   745
lemma (in complete_lattice) sup_closed [simp]:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   746
  "A \<subseteq> carrier L ==> \<Squnion>A \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   747
  by (rule supI) simp_all
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   748
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   749
lemma (in complete_lattice) top_closed [simp, intro]:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   750
  "\<top> \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   751
  by (unfold top_def) simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   752
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   753
lemma (in complete_lattice) infI:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   754
  "[| !!i. greatest L i (Lower L A) ==> P i; A \<subseteq> carrier L |]
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   755
  ==> P (\<Sqinter>A)"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   756
proof (unfold inf_def)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   757
  assume L: "A \<subseteq> carrier L"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   758
    and P: "!!l. greatest L l (Lower L A) ==> P l"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   759
  with inf_exists obtain s where "greatest L s (Lower L A)" by blast
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   760
  with L show "P (THE l. greatest L l (Lower L A))"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   761
  by (fast intro: theI2 greatest_unique P)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   762
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   763
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   764
lemma (in complete_lattice) inf_closed [simp]:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   765
  "A \<subseteq> carrier L ==> \<Sqinter>A \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   766
  by (rule infI) simp_all
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   767
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   768
lemma (in complete_lattice) bottom_closed [simp, intro]:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   769
  "\<bottom> \<in> carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   770
  by (unfold bottom_def) simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   771
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   772
text {* Jacobson: Theorem 8.1 *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   773
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   774
lemma Lower_empty [simp]:
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   775
  "Lower L {} = carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   776
  by (unfold Lower_def) simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   777
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   778
lemma Upper_empty [simp]:
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   779
  "Upper L {} = carrier L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   780
  by (unfold Upper_def) simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   781
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   782
theorem (in partial_order) complete_lattice_criterion1:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   783
  assumes top_exists: "EX g. greatest L g (carrier L)"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   784
    and inf_exists:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   785
      "!!A. [| A \<subseteq> carrier L; A ~= {} |] ==> EX i. greatest L i (Lower L A)"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   786
  shows "complete_lattice L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   787
proof (rule complete_latticeI)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   788
  from top_exists obtain top where top: "greatest L top (carrier L)" ..
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   789
  fix A
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   790
  assume L: "A \<subseteq> carrier L"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   791
  let ?B = "Upper L A"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   792
  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
   793
  then have B_non_empty: "?B ~= {}" by fast
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   794
  have B_L: "?B \<subseteq> carrier L" by simp
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   795
  from inf_exists [OF B_L B_non_empty]
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   796
  obtain b where b_inf_B: "greatest L b (Lower L ?B)" ..
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   797
  have "least L b (Upper L A)"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   798
apply (rule least_UpperI)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   799
   apply (rule greatest_le [where A = "Lower L ?B"])
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   800
    apply (rule b_inf_B)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   801
   apply (rule Lower_memI)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   802
    apply (erule UpperD)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   803
     apply assumption
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   804
    apply (rule L)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   805
   apply (fast intro: L [THEN subsetD])
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   806
  apply (erule greatest_Lower_above [OF b_inf_B])
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   807
  apply simp
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   808
 apply (rule L)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   809
apply (rule greatest_carrier [OF b_inf_B]) (* rename rule: _closed *)
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   810
done
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   811
  then show "EX s. least L s (Upper L A)" ..
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   812
next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   813
  fix A
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   814
  assume L: "A \<subseteq> carrier L"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   815
  show "EX i. greatest L i (Lower L A)"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   816
  proof (cases "A = {}")
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   817
    case True then show ?thesis
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   818
      by (simp add: top_exists)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   819
  next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   820
    case False with L show ?thesis
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   821
      by (rule inf_exists)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   822
  qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   823
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   824
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   825
(* TODO: prove dual version *)
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   826
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   827
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   828
subsection {* Examples *}
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   829
20318
0e0ea63fe768 Restructured algebra library, added ideals and quotient rings.
ballarin
parents: 19984
diff changeset
   830
subsubsection {* Powerset of a Set is a Complete Lattice *}
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   831
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   832
theorem powerset_is_complete_lattice:
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   833
  "complete_lattice (| carrier = Pow A, le = op \<subseteq> |)"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   834
  (is "complete_lattice ?L")
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   835
proof (rule partial_order.complete_latticeI)
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   836
  show "partial_order ?L"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   837
    by (rule partial_order.intro) auto
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   838
next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   839
  fix B
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   840
  assume "B \<subseteq> carrier ?L"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   841
  then have "least ?L (\<Union> B) (Upper ?L B)"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   842
    by (fastsimp intro!: least_UpperI simp: Upper_def)
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   843
  then show "EX s. least ?L s (Upper ?L B)" ..
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   844
next
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   845
  fix B
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   846
  assume "B \<subseteq> carrier ?L"
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   847
  then have "greatest ?L (\<Inter> B \<inter> A) (Lower ?L B)"
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   848
    txt {* @{term "\<Inter> B"} is not the infimum of @{term B}:
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   849
      @{term "\<Inter> {} = UNIV"} which is in general bigger than @{term "A"}! *}
22063
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   850
    by (fastsimp intro!: greatest_LowerI simp: Lower_def)
717425609192 Reverted to structure representation with records.
ballarin
parents: 21896
diff changeset
   851
  then show "EX i. greatest ?L i (Lower ?L B)" ..
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   852
qed
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   853
14751
0d7850e27fed Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents: 14706
diff changeset
   854
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
   855
  can be found in Group theory (Section~\ref{sec:subgroup-lattice}). *}
14551
2cb6ff394bfb Various changes to HOL-Algebra;
ballarin
parents:
diff changeset
   856
14693
4deda204e1d8 improved syntax;
wenzelm
parents: 14666
diff changeset
   857
end