--- a/src/HOL/Algebra/Lattice.thy Sun Oct 15 12:16:20 2006 +0200
+++ b/src/HOL/Algebra/Lattice.thy Mon Oct 16 10:27:54 2006 +0200
@@ -18,157 +18,191 @@
subsection {* Partial Orders *}
+(*
record 'a order = "'a partial_object" +
le :: "['a, 'a] => bool" (infixl "\<sqsubseteq>\<index>" 50)
+*)
-locale partial_order =
- fixes L (structure)
+text {* Locale @{text order_syntax} is required since we want to refer
+ to definitions (and their derived theorems) outside of @{text partial_order}.
+ *}
+
+locale order_syntax =
+ fixes carrier :: "'a set" and le :: "['a, 'a] => bool" (infix "\<sqsubseteq>" 50)
+
+text {* Note that the type constraints above are necessary, because the
+ definition command cannot specialise the types. *}
+
+definition (in order_syntax)
+ less (infixl "\<sqsubset>" 50) "x \<sqsubset> y == x \<sqsubseteq> y & x ~= y"
+
+text {* Upper and lower bounds of a set. *}
+
+definition (in order_syntax)
+ Upper where
+ "Upper A == {u. (ALL x. x \<in> A \<inter> carrier --> x \<sqsubseteq> u)} \<inter>
+ carrier"
+
+definition (in order_syntax)
+ Lower :: "'a set => 'a set"
+ "Lower A == {l. (ALL x. x \<in> A \<inter> carrier --> l \<sqsubseteq> x)} \<inter>
+ carrier"
+
+text {* Least and greatest, as predicate. *}
+
+definition (in order_syntax)
+ least :: "['a, 'a set] => bool"
+ "least l A == A \<subseteq> carrier & l \<in> A & (ALL x : A. l \<sqsubseteq> x)"
+
+definition (in order_syntax)
+ greatest :: "['a, 'a set] => bool"
+ "greatest g A == A \<subseteq> carrier & g \<in> A & (ALL x : A. x \<sqsubseteq> g)"
+
+text {* Supremum and infimum *}
+
+definition (in order_syntax)
+ sup :: "'a set => 'a" ("\<Squnion>") (* FIXME precedence [90] 90 *)
+ "\<Squnion>A == THE x. least x (Upper A)"
+
+definition (in order_syntax)
+ inf :: "'a set => 'a" ("\<Sqinter>") (* FIXME precedence *)
+ "\<Sqinter>A == THE x. greatest x (Lower A)"
+
+definition (in order_syntax)
+ join :: "['a, 'a] => 'a" (infixl "\<squnion>" 65)
+ "x \<squnion> y == sup {x, y}"
+
+definition (in order_syntax)
+ meet :: "['a, 'a] => 'a" (infixl "\<sqinter>" 70)
+ "x \<sqinter> y == inf {x, y}"
+
+locale partial_order = order_syntax +
assumes refl [intro, simp]:
- "x \<in> carrier L ==> x \<sqsubseteq> x"
+ "x \<in> carrier ==> x \<sqsubseteq> x"
and anti_sym [intro]:
- "[| x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> carrier L; y \<in> carrier L |] ==> x = y"
+ "[| x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> carrier; y \<in> carrier |] ==> x = y"
and trans [trans]:
"[| x \<sqsubseteq> y; y \<sqsubseteq> z;
- x \<in> carrier L; y \<in> carrier L; z \<in> carrier L |] ==> x \<sqsubseteq> z"
-
-constdefs (structure L)
- lless :: "[_, 'a, 'a] => bool" (infixl "\<sqsubset>\<index>" 50)
- "x \<sqsubset> y == x \<sqsubseteq> y & x ~= y"
-
- -- {* Upper and lower bounds of a set. *}
- Upper :: "[_, 'a set] => 'a set"
- "Upper L A == {u. (ALL x. x \<in> A \<inter> carrier L --> x \<sqsubseteq> u)} \<inter>
- carrier L"
-
- Lower :: "[_, 'a set] => 'a set"
- "Lower L A == {l. (ALL x. x \<in> A \<inter> carrier L --> l \<sqsubseteq> x)} \<inter>
- carrier L"
+ x \<in> carrier; y \<in> carrier; z \<in> carrier |] ==> x \<sqsubseteq> z"
- -- {* Least and greatest, as predicate. *}
- least :: "[_, 'a, 'a set] => bool"
- "least L l A == A \<subseteq> carrier L & l \<in> A & (ALL x : A. l \<sqsubseteq> x)"
-
- greatest :: "[_, 'a, 'a set] => bool"
- "greatest L g A == A \<subseteq> carrier L & g \<in> A & (ALL x : A. x \<sqsubseteq> g)"
-
- -- {* Supremum and infimum *}
- sup :: "[_, 'a set] => 'a" ("\<Squnion>\<index>_" [90] 90)
- "\<Squnion>A == THE x. least L x (Upper L A)"
-
- inf :: "[_, 'a set] => 'a" ("\<Sqinter>\<index>_" [90] 90)
- "\<Sqinter>A == THE x. greatest L x (Lower L A)"
-
- join :: "[_, 'a, 'a] => 'a" (infixl "\<squnion>\<index>" 65)
- "x \<squnion> y == sup L {x, y}"
-
- meet :: "[_, 'a, 'a] => 'a" (infixl "\<sqinter>\<index>" 70)
- "x \<sqinter> y == inf L {x, y}"
+abbreviation (in partial_order)
+ less (infixl "\<sqsubset>" 50) "less == order_syntax.less le"
+abbreviation (in partial_order)
+ Upper where "Upper == order_syntax.Upper carrier le"
+abbreviation (in partial_order)
+ Lower where "Lower == order_syntax.Lower carrier le"
+abbreviation (in partial_order)
+ least where "least == order_syntax.least carrier le"
+abbreviation (in partial_order)
+ greatest where "greatest == order_syntax.greatest carrier le"
+abbreviation (in partial_order)
+ sup ("\<Squnion>") (* FIXME precedence *) "sup == order_syntax.sup carrier le"
+abbreviation (in partial_order)
+ inf ("\<Sqinter>") (* FIXME precedence *) "inf == order_syntax.inf carrier le"
+abbreviation (in partial_order)
+ join (infixl "\<squnion>" 65) "join == order_syntax.join carrier le"
+abbreviation (in partial_order)
+ meet (infixl "\<sqinter>" 70) "meet == order_syntax.meet carrier le"
subsubsection {* Upper *}
-lemma Upper_closed [intro, simp]:
- "Upper L A \<subseteq> carrier L"
+lemma (in order_syntax) Upper_closed [intro, simp]:
+ "Upper A \<subseteq> carrier"
by (unfold Upper_def) clarify
-lemma UpperD [dest]:
+lemma (in order_syntax) UpperD [dest]:
fixes L (structure)
- shows "[| u \<in> Upper L A; x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> u"
+ shows "[| u \<in> Upper A; x \<in> A; A \<subseteq> carrier |] ==> x \<sqsubseteq> u"
by (unfold Upper_def) blast
-lemma Upper_memI:
+lemma (in order_syntax) Upper_memI:
fixes L (structure)
- shows "[| !! y. y \<in> A ==> y \<sqsubseteq> x; x \<in> carrier L |] ==> x \<in> Upper L A"
+ shows "[| !! y. y \<in> A ==> y \<sqsubseteq> x; x \<in> carrier |] ==> x \<in> Upper A"
by (unfold Upper_def) blast
-lemma Upper_antimono:
- "A \<subseteq> B ==> Upper L B \<subseteq> Upper L A"
+lemma (in order_syntax) Upper_antimono:
+ "A \<subseteq> B ==> Upper B \<subseteq> Upper A"
by (unfold Upper_def) blast
subsubsection {* Lower *}
-lemma Lower_closed [intro, simp]:
- "Lower L A \<subseteq> carrier L"
+lemma (in order_syntax) Lower_closed [intro, simp]:
+ "Lower A \<subseteq> carrier"
by (unfold Lower_def) clarify
-lemma LowerD [dest]:
- fixes L (structure)
- shows "[| l \<in> Lower L A; x \<in> A; A \<subseteq> carrier L |] ==> l \<sqsubseteq> x"
+lemma (in order_syntax) LowerD [dest]:
+ "[| l \<in> Lower A; x \<in> A; A \<subseteq> carrier |] ==> l \<sqsubseteq> x"
by (unfold Lower_def) blast
-lemma Lower_memI:
- fixes L (structure)
- shows "[| !! y. y \<in> A ==> x \<sqsubseteq> y; x \<in> carrier L |] ==> x \<in> Lower L A"
+lemma (in order_syntax) Lower_memI:
+ "[| !! y. y \<in> A ==> x \<sqsubseteq> y; x \<in> carrier |] ==> x \<in> Lower A"
by (unfold Lower_def) blast
-lemma Lower_antimono:
- "A \<subseteq> B ==> Lower L B \<subseteq> Lower L A"
+lemma (in order_syntax) Lower_antimono:
+ "A \<subseteq> B ==> Lower B \<subseteq> Lower A"
by (unfold Lower_def) blast
subsubsection {* least *}
-lemma least_carrier [intro, simp]:
- shows "least L l A ==> l \<in> carrier L"
+lemma (in order_syntax) least_carrier [intro, simp]:
+ "least l A ==> l \<in> carrier"
by (unfold least_def) fast
-lemma least_mem:
- "least L l A ==> l \<in> A"
+lemma (in order_syntax) least_mem:
+ "least l A ==> l \<in> A"
by (unfold least_def) fast
lemma (in partial_order) least_unique:
- "[| least L x A; least L y A |] ==> x = y"
+ "[| least x A; least y A |] ==> x = y"
by (unfold least_def) blast
-lemma least_le:
- fixes L (structure)
- shows "[| least L x A; a \<in> A |] ==> x \<sqsubseteq> a"
+lemma (in order_syntax) least_le:
+ "[| least x A; a \<in> A |] ==> x \<sqsubseteq> a"
by (unfold least_def) fast
-lemma least_UpperI:
- fixes L (structure)
+lemma (in order_syntax) least_UpperI:
assumes above: "!! x. x \<in> A ==> x \<sqsubseteq> s"
- and below: "!! y. y \<in> Upper L A ==> s \<sqsubseteq> y"
- and L: "A \<subseteq> carrier L" "s \<in> carrier L"
- shows "least L s (Upper L A)"
+ and below: "!! y. y \<in> Upper A ==> s \<sqsubseteq> y"
+ and L: "A \<subseteq> carrier" "s \<in> carrier"
+ shows "least s (Upper A)"
proof -
- have "Upper L A \<subseteq> carrier L" by simp
- moreover from above L have "s \<in> Upper L A" by (simp add: Upper_def)
- moreover from below have "ALL x : Upper L A. s \<sqsubseteq> x" by fast
+ have "Upper A \<subseteq> carrier" by simp
+ moreover from above L have "s \<in> Upper A" by (simp add: Upper_def)
+ moreover from below have "ALL x : Upper A. s \<sqsubseteq> x" by fast
ultimately show ?thesis by (simp add: least_def)
qed
subsubsection {* greatest *}
-lemma greatest_carrier [intro, simp]:
- shows "greatest L l A ==> l \<in> carrier L"
+lemma (in order_syntax) greatest_carrier [intro, simp]:
+ "greatest l A ==> l \<in> carrier"
by (unfold greatest_def) fast
-lemma greatest_mem:
- "greatest L l A ==> l \<in> A"
+lemma (in order_syntax) greatest_mem:
+ "greatest l A ==> l \<in> A"
by (unfold greatest_def) fast
lemma (in partial_order) greatest_unique:
- "[| greatest L x A; greatest L y A |] ==> x = y"
+ "[| greatest x A; greatest y A |] ==> x = y"
by (unfold greatest_def) blast
-lemma greatest_le:
- fixes L (structure)
- shows "[| greatest L x A; a \<in> A |] ==> a \<sqsubseteq> x"
+lemma (in order_syntax) greatest_le:
+ "[| greatest x A; a \<in> A |] ==> a \<sqsubseteq> x"
by (unfold greatest_def) fast
-lemma greatest_LowerI:
- fixes L (structure)
+lemma (in order_syntax) greatest_LowerI:
assumes below: "!! x. x \<in> A ==> i \<sqsubseteq> x"
- and above: "!! y. y \<in> Lower L A ==> y \<sqsubseteq> i"
- and L: "A \<subseteq> carrier L" "i \<in> carrier L"
- shows "greatest L i (Lower L A)"
+ and above: "!! y. y \<in> Lower A ==> y \<sqsubseteq> i"
+ and L: "A \<subseteq> carrier" "i \<in> carrier"
+ shows "greatest i (Lower A)"
proof -
- have "Lower L A \<subseteq> carrier L" by simp
- moreover from below L have "i \<in> Lower L A" by (simp add: Lower_def)
- moreover from above have "ALL x : Lower L A. x \<sqsubseteq> i" by fast
+ have "Lower A \<subseteq> carrier" by simp
+ moreover from below L have "i \<in> Lower A" by (simp add: Lower_def)
+ moreover from above have "ALL x : Lower A. x \<sqsubseteq> i" by fast
ultimately show ?thesis by (simp add: greatest_def)
qed
@@ -177,63 +211,80 @@
locale lattice = partial_order +
assumes sup_of_two_exists:
- "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. least L s (Upper L {x, y})"
+ "[| x \<in> carrier; y \<in> carrier |] ==> EX s. order_syntax.least carrier le s (order_syntax.Upper carrier le {x, y})"
and inf_of_two_exists:
- "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. greatest L s (Lower L {x, y})"
+ "[| x \<in> carrier; y \<in> carrier |] ==> EX s. order_syntax.greatest carrier le s (order_syntax.Lower carrier le {x, y})"
-lemma least_Upper_above:
- fixes L (structure)
- shows "[| least L s (Upper L A); x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> s"
+abbreviation (in lattice)
+ less (infixl "\<sqsubset>" 50) "less == order_syntax.less le"
+abbreviation (in lattice)
+ Upper where "Upper == order_syntax.Upper carrier le"
+abbreviation (in lattice)
+ Lower where "Lower == order_syntax.Lower carrier le"
+abbreviation (in lattice)
+ least where "least == order_syntax.least carrier le"
+abbreviation (in lattice)
+ greatest where "greatest == order_syntax.greatest carrier le"
+abbreviation (in lattice)
+ sup ("\<Squnion>") (* FIXME precedence *) "sup == order_syntax.sup carrier le"
+abbreviation (in lattice)
+ inf ("\<Sqinter>") (* FIXME precedence *) "inf == order_syntax.inf carrier le"
+abbreviation (in lattice)
+ join (infixl "\<squnion>" 65) "join == order_syntax.join carrier le"
+abbreviation (in lattice)
+ meet (infixl "\<sqinter>" 70) "meet == order_syntax.meet carrier le"
+
+lemma (in order_syntax) least_Upper_above:
+ "[| least s (Upper A); x \<in> A; A \<subseteq> carrier |] ==> x \<sqsubseteq> s"
by (unfold least_def) blast
-lemma greatest_Lower_above:
- fixes L (structure)
- shows "[| greatest L i (Lower L A); x \<in> A; A \<subseteq> carrier L |] ==> i \<sqsubseteq> x"
+lemma (in order_syntax) greatest_Lower_above:
+ "[| greatest i (Lower A); x \<in> A; A \<subseteq> carrier |] ==> i \<sqsubseteq> x"
by (unfold greatest_def) blast
subsubsection {* Supremum *}
lemma (in lattice) joinI:
- "[| !!l. least L l (Upper L {x, y}) ==> P l; x \<in> carrier L; y \<in> carrier L |]
+ "[| !!l. least l (Upper {x, y}) ==> P l; x \<in> carrier; y \<in> carrier |]
==> P (x \<squnion> y)"
proof (unfold join_def sup_def)
- assume L: "x \<in> carrier L" "y \<in> carrier L"
- and P: "!!l. least L l (Upper L {x, y}) ==> P l"
- with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
- with L show "P (THE l. least L l (Upper L {x, y}))"
+ assume L: "x \<in> carrier" "y \<in> carrier"
+ and P: "!!l. least l (Upper {x, y}) ==> P l"
+ with sup_of_two_exists obtain s where "least s (Upper {x, y})" by fast
+ with L show "P (THE l. least l (Upper {x, y}))"
by (fast intro: theI2 least_unique P)
qed
lemma (in lattice) join_closed [simp]:
- "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<squnion> y \<in> carrier L"
+ "[| x \<in> carrier; y \<in> carrier |] ==> x \<squnion> y \<in> carrier"
by (rule joinI) (rule least_carrier)
-lemma (in partial_order) sup_of_singletonI: (* only reflexivity needed ? *)
- "x \<in> carrier L ==> least L x (Upper L {x})"
+lemma (in partial_order) sup_of_singletonI: (* only reflexivity needed ? *)
+ "x \<in> carrier ==> least x (Upper {x})"
by (rule least_UpperI) fast+
lemma (in partial_order) sup_of_singleton [simp]:
- "x \<in> carrier L ==> \<Squnion>{x} = x"
+ "x \<in> carrier ==> \<Squnion>{x} = x"
by (unfold sup_def) (blast intro: least_unique least_UpperI sup_of_singletonI)
text {* Condition on @{text A}: supremum exists. *}
lemma (in lattice) sup_insertI:
- "[| !!s. least L s (Upper L (insert x A)) ==> P s;
- least L a (Upper L A); x \<in> carrier L; A \<subseteq> carrier L |]
+ "[| !!s. least s (Upper (insert x A)) ==> P s;
+ least a (Upper A); x \<in> carrier; A \<subseteq> carrier |]
==> P (\<Squnion>(insert x A))"
proof (unfold sup_def)
- assume L: "x \<in> carrier L" "A \<subseteq> carrier L"
- and P: "!!l. least L l (Upper L (insert x A)) ==> P l"
- and least_a: "least L a (Upper L A)"
- from L least_a have La: "a \<in> carrier L" by simp
+ assume L: "x \<in> carrier" "A \<subseteq> carrier"
+ and P: "!!l. least l (Upper (insert x A)) ==> P l"
+ and least_a: "least a (Upper A)"
+ from least_a have La: "a \<in> carrier" by simp
from L sup_of_two_exists least_a
- obtain s where least_s: "least L s (Upper L {a, x})" by blast
- show "P (THE l. least L l (Upper L (insert x A)))"
+ obtain s where least_s: "least s (Upper {a, x})" by blast
+ show "P (THE l. least l (Upper (insert x A)))"
proof (rule theI2)
- show "least L s (Upper L (insert x A))"
+ show "least s (Upper (insert x A))"
proof (rule least_UpperI)
fix z
assume "z \<in> insert x A"
@@ -248,15 +299,15 @@
qed
next
fix y
- assume y: "y \<in> Upper L (insert x A)"
+ assume y: "y \<in> Upper (insert x A)"
show "s \<sqsubseteq> y"
proof (rule least_le [OF least_s], rule Upper_memI)
fix z
assume z: "z \<in> {a, x}"
then show "z \<sqsubseteq> y"
proof
- have y': "y \<in> Upper L A"
- apply (rule subsetD [where A = "Upper L (insert x A)"])
+ have y': "y \<in> Upper A"
+ apply (rule subsetD [where A = "Upper (insert x A)"])
apply (rule Upper_antimono) apply clarify apply assumption
done
assume "z = a"
@@ -267,15 +318,15 @@
qed
qed (rule Upper_closed [THEN subsetD])
next
- from L show "insert x A \<subseteq> carrier L" by simp
- from least_s show "s \<in> carrier L" by simp
+ from L show "insert x A \<subseteq> carrier" by simp
+ from least_s show "s \<in> carrier" by simp
qed
next
fix l
- assume least_l: "least L l (Upper L (insert x A))"
+ assume least_l: "least l (Upper (insert x A))"
show "l = s"
proof (rule least_unique)
- show "least L s (Upper L (insert x A))"
+ show "least s (Upper (insert x A))"
proof (rule least_UpperI)
fix z
assume "z \<in> insert x A"
@@ -290,15 +341,15 @@
qed
next
fix y
- assume y: "y \<in> Upper L (insert x A)"
+ assume y: "y \<in> Upper (insert x A)"
show "s \<sqsubseteq> y"
proof (rule least_le [OF least_s], rule Upper_memI)
fix z
assume z: "z \<in> {a, x}"
then show "z \<sqsubseteq> y"
proof
- have y': "y \<in> Upper L A"
- apply (rule subsetD [where A = "Upper L (insert x A)"])
+ have y': "y \<in> Upper A"
+ apply (rule subsetD [where A = "Upper (insert x A)"])
apply (rule Upper_antimono) apply clarify apply assumption
done
assume "z = a"
@@ -309,15 +360,15 @@
qed
qed (rule Upper_closed [THEN subsetD])
next
- from L show "insert x A \<subseteq> carrier L" by simp
- from least_s show "s \<in> carrier L" by simp
+ from L show "insert x A \<subseteq> carrier" by simp
+ from least_s show "s \<in> carrier" by simp
qed
qed
qed
qed
lemma (in lattice) finite_sup_least:
- "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> least L (\<Squnion>A) (Upper L A)"
+ "[| finite A; A \<subseteq> carrier; A ~= {} |] ==> least (\<Squnion>A) (Upper A)"
proof (induct set: Finites)
case empty
then show ?case by simp
@@ -329,15 +380,15 @@
with insert show ?thesis by (simp add: sup_of_singletonI)
next
case False
- with insert have "least L (\<Squnion>A) (Upper L A)" by simp
+ with insert have "least (\<Squnion>A) (Upper A)" by simp
with _ show ?thesis
by (rule sup_insertI) (simp_all add: insert [simplified])
qed
qed
lemma (in lattice) finite_sup_insertI:
- assumes P: "!!l. least L l (Upper L (insert x A)) ==> P l"
- and xA: "finite A" "x \<in> carrier L" "A \<subseteq> carrier L"
+ assumes P: "!!l. least l (Upper (insert x A)) ==> P l"
+ and xA: "finite A" "x \<in> carrier" "A \<subseteq> carrier"
shows "P (\<Squnion> (insert x A))"
proof (cases "A = {}")
case True with P and xA show ?thesis
@@ -348,7 +399,7 @@
qed
lemma (in lattice) finite_sup_closed:
- "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Squnion>A \<in> carrier L"
+ "[| finite A; A \<subseteq> carrier; A ~= {} |] ==> \<Squnion>A \<in> carrier"
proof (induct set: Finites)
case empty then show ?case by simp
next
@@ -357,39 +408,39 @@
qed
lemma (in lattice) join_left:
- "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> x \<squnion> y"
+ "[| x \<in> carrier; y \<in> carrier |] ==> x \<sqsubseteq> x \<squnion> y"
by (rule joinI [folded join_def]) (blast dest: least_mem)
lemma (in lattice) join_right:
- "[| x \<in> carrier L; y \<in> carrier L |] ==> y \<sqsubseteq> x \<squnion> y"
+ "[| x \<in> carrier; y \<in> carrier |] ==> y \<sqsubseteq> x \<squnion> y"
by (rule joinI [folded join_def]) (blast dest: least_mem)
lemma (in lattice) sup_of_two_least:
- "[| x \<in> carrier L; y \<in> carrier L |] ==> least L (\<Squnion>{x, y}) (Upper L {x, y})"
+ "[| x \<in> carrier; y \<in> carrier |] ==> least (\<Squnion>{x, y}) (Upper {x, y})"
proof (unfold sup_def)
- assume L: "x \<in> carrier L" "y \<in> carrier L"
- with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
- with L show "least L (THE xa. least L xa (Upper L {x, y})) (Upper L {x, y})"
+ assume L: "x \<in> carrier" "y \<in> carrier"
+ with sup_of_two_exists obtain s where "least s (Upper {x, y})" by fast
+ with L show "least (THE xa. least xa (Upper {x, y})) (Upper {x, y})"
by (fast intro: theI2 least_unique) (* blast fails *)
qed
lemma (in lattice) join_le:
assumes sub: "x \<sqsubseteq> z" "y \<sqsubseteq> z"
- and L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
+ and L: "x \<in> carrier" "y \<in> carrier" "z \<in> carrier"
shows "x \<squnion> y \<sqsubseteq> z"
proof (rule joinI)
fix s
- assume "least L s (Upper L {x, y})"
+ assume "least s (Upper {x, y})"
with sub L show "s \<sqsubseteq> z" by (fast elim: least_le intro: Upper_memI)
qed
lemma (in lattice) join_assoc_lemma:
- assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
+ assumes L: "x \<in> carrier" "y \<in> carrier" "z \<in> carrier"
shows "x \<squnion> (y \<squnion> z) = \<Squnion>{x, y, z}"
proof (rule finite_sup_insertI)
-- {* The textbook argument in Jacobson I, p 457 *}
fix s
- assume sup: "least L s (Upper L {x, y, z})"
+ assume sup: "least s (Upper {x, y, z})"
show "x \<squnion> (y \<squnion> z) = s"
proof (rule anti_sym)
from sup L show "x \<squnion> (y \<squnion> z) \<sqsubseteq> s"
@@ -401,13 +452,12 @@
qed (simp_all add: L least_carrier [OF sup])
qed (simp_all add: L)
-lemma join_comm:
- fixes L (structure)
- shows "x \<squnion> y = y \<squnion> x"
+lemma (in order_syntax) join_comm:
+ "x \<squnion> y = y \<squnion> x"
by (unfold join_def) (simp add: insert_commute)
lemma (in lattice) join_assoc:
- assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
+ assumes L: "x \<in> carrier" "y \<in> carrier" "z \<in> carrier"
shows "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
proof -
have "(x \<squnion> y) \<squnion> z = z \<squnion> (x \<squnion> y)" by (simp only: join_comm)
@@ -421,45 +471,45 @@
subsubsection {* Infimum *}
lemma (in lattice) meetI:
- "[| !!i. greatest L i (Lower L {x, y}) ==> P i;
- x \<in> carrier L; y \<in> carrier L |]
+ "[| !!i. greatest i (Lower {x, y}) ==> P i;
+ x \<in> carrier; y \<in> carrier |]
==> P (x \<sqinter> y)"
proof (unfold meet_def inf_def)
- assume L: "x \<in> carrier L" "y \<in> carrier L"
- and P: "!!g. greatest L g (Lower L {x, y}) ==> P g"
- with inf_of_two_exists obtain i where "greatest L i (Lower L {x, y})" by fast
- with L show "P (THE g. greatest L g (Lower L {x, y}))"
+ assume L: "x \<in> carrier" "y \<in> carrier"
+ and P: "!!g. greatest g (Lower {x, y}) ==> P g"
+ with inf_of_two_exists obtain i where "greatest i (Lower {x, y})" by fast
+ with L show "P (THE g. greatest g (Lower {x, y}))"
by (fast intro: theI2 greatest_unique P)
qed
lemma (in lattice) meet_closed [simp]:
- "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<in> carrier L"
+ "[| x \<in> carrier; y \<in> carrier |] ==> x \<sqinter> y \<in> carrier"
by (rule meetI) (rule greatest_carrier)
lemma (in partial_order) inf_of_singletonI: (* only reflexivity needed ? *)
- "x \<in> carrier L ==> greatest L x (Lower L {x})"
+ "x \<in> carrier ==> greatest x (Lower {x})"
by (rule greatest_LowerI) fast+
lemma (in partial_order) inf_of_singleton [simp]:
- "x \<in> carrier L ==> \<Sqinter> {x} = x"
+ "x \<in> carrier ==> \<Sqinter> {x} = x"
by (unfold inf_def) (blast intro: greatest_unique greatest_LowerI inf_of_singletonI)
text {* Condition on A: infimum exists. *}
lemma (in lattice) inf_insertI:
- "[| !!i. greatest L i (Lower L (insert x A)) ==> P i;
- greatest L a (Lower L A); x \<in> carrier L; A \<subseteq> carrier L |]
+ "[| !!i. greatest i (Lower (insert x A)) ==> P i;
+ greatest a (Lower A); x \<in> carrier; A \<subseteq> carrier |]
==> P (\<Sqinter>(insert x A))"
proof (unfold inf_def)
- assume L: "x \<in> carrier L" "A \<subseteq> carrier L"
- and P: "!!g. greatest L g (Lower L (insert x A)) ==> P g"
- and greatest_a: "greatest L a (Lower L A)"
- from L greatest_a have La: "a \<in> carrier L" by simp
+ assume L: "x \<in> carrier" "A \<subseteq> carrier"
+ and P: "!!g. greatest g (Lower (insert x A)) ==> P g"
+ and greatest_a: "greatest a (Lower A)"
+ from greatest_a have La: "a \<in> carrier" by simp
from L inf_of_two_exists greatest_a
- obtain i where greatest_i: "greatest L i (Lower L {a, x})" by blast
- show "P (THE g. greatest L g (Lower L (insert x A)))"
+ obtain i where greatest_i: "greatest i (Lower {a, x})" by blast
+ show "P (THE g. greatest g (Lower (insert x A)))"
proof (rule theI2)
- show "greatest L i (Lower L (insert x A))"
+ show "greatest i (Lower (insert x A))"
proof (rule greatest_LowerI)
fix z
assume "z \<in> insert x A"
@@ -474,15 +524,15 @@
qed
next
fix y
- assume y: "y \<in> Lower L (insert x A)"
+ assume y: "y \<in> Lower (insert x A)"
show "y \<sqsubseteq> i"
proof (rule greatest_le [OF greatest_i], rule Lower_memI)
fix z
assume z: "z \<in> {a, x}"
then show "y \<sqsubseteq> z"
proof
- have y': "y \<in> Lower L A"
- apply (rule subsetD [where A = "Lower L (insert x A)"])
+ have y': "y \<in> Lower A"
+ apply (rule subsetD [where A = "Lower (insert x A)"])
apply (rule Lower_antimono) apply clarify apply assumption
done
assume "z = a"
@@ -493,15 +543,15 @@
qed
qed (rule Lower_closed [THEN subsetD])
next
- from L show "insert x A \<subseteq> carrier L" by simp
- from greatest_i show "i \<in> carrier L" by simp
+ from L show "insert x A \<subseteq> carrier" by simp
+ from greatest_i show "i \<in> carrier" by simp
qed
next
fix g
- assume greatest_g: "greatest L g (Lower L (insert x A))"
+ assume greatest_g: "greatest g (Lower (insert x A))"
show "g = i"
proof (rule greatest_unique)
- show "greatest L i (Lower L (insert x A))"
+ show "greatest i (Lower (insert x A))"
proof (rule greatest_LowerI)
fix z
assume "z \<in> insert x A"
@@ -516,15 +566,15 @@
qed
next
fix y
- assume y: "y \<in> Lower L (insert x A)"
+ assume y: "y \<in> Lower (insert x A)"
show "y \<sqsubseteq> i"
proof (rule greatest_le [OF greatest_i], rule Lower_memI)
fix z
assume z: "z \<in> {a, x}"
then show "y \<sqsubseteq> z"
proof
- have y': "y \<in> Lower L A"
- apply (rule subsetD [where A = "Lower L (insert x A)"])
+ have y': "y \<in> Lower A"
+ apply (rule subsetD [where A = "Lower (insert x A)"])
apply (rule Lower_antimono) apply clarify apply assumption
done
assume "z = a"
@@ -535,15 +585,15 @@
qed
qed (rule Lower_closed [THEN subsetD])
next
- from L show "insert x A \<subseteq> carrier L" by simp
- from greatest_i show "i \<in> carrier L" by simp
+ from L show "insert x A \<subseteq> carrier" by simp
+ from greatest_i show "i \<in> carrier" by simp
qed
qed
qed
qed
lemma (in lattice) finite_inf_greatest:
- "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> greatest L (\<Sqinter>A) (Lower L A)"
+ "[| finite A; A \<subseteq> carrier; A ~= {} |] ==> greatest (\<Sqinter>A) (Lower A)"
proof (induct set: Finites)
case empty then show ?case by simp
next
@@ -556,14 +606,14 @@
case False
from insert show ?thesis
proof (rule_tac inf_insertI)
- from False insert show "greatest L (\<Sqinter>A) (Lower L A)" by simp
+ from False insert show "greatest (\<Sqinter>A) (Lower A)" by simp
qed simp_all
qed
qed
lemma (in lattice) finite_inf_insertI:
- assumes P: "!!i. greatest L i (Lower L (insert x A)) ==> P i"
- and xA: "finite A" "x \<in> carrier L" "A \<subseteq> carrier L"
+ assumes P: "!!i. greatest i (Lower (insert x A)) ==> P i"
+ and xA: "finite A" "x \<in> carrier" "A \<subseteq> carrier"
shows "P (\<Sqinter> (insert x A))"
proof (cases "A = {}")
case True with P and xA show ?thesis
@@ -574,7 +624,7 @@
qed
lemma (in lattice) finite_inf_closed:
- "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Sqinter>A \<in> carrier L"
+ "[| finite A; A \<subseteq> carrier; A ~= {} |] ==> \<Sqinter>A \<in> carrier"
proof (induct set: Finites)
case empty then show ?case by simp
next
@@ -583,41 +633,41 @@
qed
lemma (in lattice) meet_left:
- "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> x"
+ "[| x \<in> carrier; y \<in> carrier |] ==> x \<sqinter> y \<sqsubseteq> x"
by (rule meetI [folded meet_def]) (blast dest: greatest_mem)
lemma (in lattice) meet_right:
- "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> y"
+ "[| x \<in> carrier; y \<in> carrier |] ==> x \<sqinter> y \<sqsubseteq> y"
by (rule meetI [folded meet_def]) (blast dest: greatest_mem)
lemma (in lattice) inf_of_two_greatest:
- "[| x \<in> carrier L; y \<in> carrier L |] ==>
- greatest L (\<Sqinter> {x, y}) (Lower L {x, y})"
+ "[| x \<in> carrier; y \<in> carrier |] ==>
+ greatest (\<Sqinter> {x, y}) (Lower {x, y})"
proof (unfold inf_def)
- assume L: "x \<in> carrier L" "y \<in> carrier L"
- with inf_of_two_exists obtain s where "greatest L s (Lower L {x, y})" by fast
+ assume L: "x \<in> carrier" "y \<in> carrier"
+ with inf_of_two_exists obtain s where "greatest s (Lower {x, y})" by fast
with L
- show "greatest L (THE xa. greatest L xa (Lower L {x, y})) (Lower L {x, y})"
+ show "greatest (THE xa. greatest xa (Lower {x, y})) (Lower {x, y})"
by (fast intro: theI2 greatest_unique) (* blast fails *)
qed
lemma (in lattice) meet_le:
assumes sub: "z \<sqsubseteq> x" "z \<sqsubseteq> y"
- and L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
+ and L: "x \<in> carrier" "y \<in> carrier" "z \<in> carrier"
shows "z \<sqsubseteq> x \<sqinter> y"
proof (rule meetI)
fix i
- assume "greatest L i (Lower L {x, y})"
+ assume "greatest i (Lower {x, y})"
with sub L show "z \<sqsubseteq> i" by (fast elim: greatest_le intro: Lower_memI)
qed
lemma (in lattice) meet_assoc_lemma:
- assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
+ assumes L: "x \<in> carrier" "y \<in> carrier" "z \<in> carrier"
shows "x \<sqinter> (y \<sqinter> z) = \<Sqinter>{x, y, z}"
proof (rule finite_inf_insertI)
txt {* The textbook argument in Jacobson I, p 457 *}
fix i
- assume inf: "greatest L i (Lower L {x, y, z})"
+ assume inf: "greatest i (Lower {x, y, z})"
show "x \<sqinter> (y \<sqinter> z) = i"
proof (rule anti_sym)
from inf L show "i \<sqsubseteq> x \<sqinter> (y \<sqinter> z)"
@@ -629,13 +679,12 @@
qed (simp_all add: L greatest_carrier [OF inf])
qed (simp_all add: L)
-lemma meet_comm:
- fixes L (structure)
- shows "x \<sqinter> y = y \<sqinter> x"
+lemma (in order_syntax) meet_comm:
+ "x \<sqinter> y = y \<sqinter> x"
by (unfold meet_def) (simp add: insert_commute)
lemma (in lattice) meet_assoc:
- assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
+ assumes L: "x \<in> carrier" "y \<in> carrier" "z \<in> carrier"
shows "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
proof -
have "(x \<sqinter> y) \<sqinter> z = z \<sqinter> (x \<sqinter> y)" by (simp only: meet_comm)
@@ -649,51 +698,70 @@
subsection {* Total Orders *}
locale total_order = lattice +
- assumes total: "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
+ assumes total: "[| x \<in> carrier; y \<in> carrier |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
+
+abbreviation (in total_order)
+ less (infixl "\<sqsubset>" 50) "less == order_syntax.less le"
+abbreviation (in total_order)
+ Upper where "Upper == order_syntax.Upper carrier le"
+abbreviation (in total_order)
+ Lower where "Lower == order_syntax.Lower carrier le"
+abbreviation (in total_order)
+ least where "least == order_syntax.least carrier le"
+abbreviation (in total_order)
+ greatest where "greatest == order_syntax.greatest carrier le"
+abbreviation (in total_order)
+ sup ("\<Squnion>") (* FIXME precedence *) "sup == order_syntax.sup carrier le"
+abbreviation (in total_order)
+ inf ("\<Sqinter>") (* FIXME precedence *) "inf == order_syntax.inf carrier le"
+abbreviation (in total_order)
+ join (infixl "\<squnion>" 65) "join == order_syntax.join carrier le"
+abbreviation (in total_order)
+ meet (infixl "\<sqinter>" 70) "meet == order_syntax.meet carrier le"
text {* Introduction rule: the usual definition of total order *}
lemma (in partial_order) total_orderI:
- assumes total: "!!x y. [| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
- shows "total_order L"
+ assumes total: "!!x y. [| x \<in> carrier; y \<in> carrier |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x"
+ shows "total_order carrier le"
proof intro_locales
- show "lattice_axioms L"
+ show "lattice_axioms carrier le"
proof (rule lattice_axioms.intro)
fix x y
- assume L: "x \<in> carrier L" "y \<in> carrier L"
- show "EX s. least L s (Upper L {x, y})"
+ assume L: "x \<in> carrier" "y \<in> carrier"
+ show "EX s. least s (Upper {x, y})"
proof -
note total L
moreover
{
assume "x \<sqsubseteq> y"
- with L have "least L y (Upper L {x, y})"
+ with L have "least y (Upper {x, y})"
by (rule_tac least_UpperI) auto
}
moreover
{
assume "y \<sqsubseteq> x"
- with L have "least L x (Upper L {x, y})"
+ with L have "least x (Upper {x, y})"
by (rule_tac least_UpperI) auto
}
ultimately show ?thesis by blast
qed
next
fix x y
- assume L: "x \<in> carrier L" "y \<in> carrier L"
- show "EX i. greatest L i (Lower L {x, y})"
+ assume L: "x \<in> carrier" "y \<in> carrier"
+ show "EX i. greatest i (Lower {x, y})"
proof -
note total L
moreover
{
assume "y \<sqsubseteq> x"
- with L have "greatest L y (Lower L {x, y})"
+ with L have "greatest y (Lower {x, y})"
by (rule_tac greatest_LowerI) auto
}
moreover
{
assume "x \<sqsubseteq> y"
- with L have "greatest L x (Lower L {x, y})"
+ with L have "greatest x (Lower {x, y})"
by (rule_tac greatest_LowerI) auto
}
ultimately show ?thesis by blast
@@ -706,97 +774,134 @@
locale complete_lattice = lattice +
assumes sup_exists:
- "[| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)"
+ "[| A \<subseteq> carrier |] ==> EX s. order_syntax.least carrier le s (order_syntax.Upper carrier le A)"
and inf_exists:
- "[| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)"
+ "[| A \<subseteq> carrier |] ==> EX i. order_syntax.greatest carrier le i (order_syntax.Lower carrier le A)"
+
+abbreviation (in complete_lattice)
+ less (infixl "\<sqsubset>" 50) "less == order_syntax.less le"
+abbreviation (in complete_lattice)
+ Upper where "Upper == order_syntax.Upper carrier le"
+abbreviation (in complete_lattice)
+ Lower where "Lower == order_syntax.Lower carrier le"
+abbreviation (in complete_lattice)
+ least where "least == order_syntax.least carrier le"
+abbreviation (in complete_lattice)
+ greatest where "greatest == order_syntax.greatest carrier le"
+abbreviation (in complete_lattice)
+ sup ("\<Squnion>") (* FIXME precedence *) "sup == order_syntax.sup carrier le"
+abbreviation (in complete_lattice)
+ inf ("\<Sqinter>") (* FIXME precedence *) "inf == order_syntax.inf carrier le"
+abbreviation (in complete_lattice)
+ join (infixl "\<squnion>" 65) "join == order_syntax.join carrier le"
+abbreviation (in complete_lattice)
+ meet (infixl "\<sqinter>" 70) "meet == order_syntax.meet carrier le"
text {* Introduction rule: the usual definition of complete lattice *}
lemma (in partial_order) complete_latticeI:
assumes sup_exists:
- "!!A. [| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)"
+ "!!A. [| A \<subseteq> carrier |] ==> EX s. least s (Upper A)"
and inf_exists:
- "!!A. [| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)"
- shows "complete_lattice L"
+ "!!A. [| A \<subseteq> carrier |] ==> EX i. greatest i (Lower A)"
+ shows "complete_lattice carrier le"
proof intro_locales
- show "lattice_axioms L"
+ show "lattice_axioms carrier le"
by (rule lattice_axioms.intro) (blast intro: sup_exists inf_exists)+
qed (assumption | rule complete_lattice_axioms.intro)+
-constdefs (structure L)
- top :: "_ => 'a" ("\<top>\<index>")
- "\<top> == sup L (carrier L)"
+definition (in order_syntax)
+ top ("\<top>")
+ "\<top> == sup carrier"
+
+definition (in order_syntax)
+ bottom ("\<bottom>")
+ "\<bottom> == inf carrier"
- bottom :: "_ => 'a" ("\<bottom>\<index>")
- "\<bottom> == inf L (carrier L)"
+abbreviation (in partial_order)
+ top ("\<top>") "top == order_syntax.top carrier le"
+abbreviation (in partial_order)
+ bottom ("\<bottom>") "bottom == order_syntax.bottom carrier le"
+abbreviation (in lattice)
+ top ("\<top>") "top == order_syntax.top carrier le"
+abbreviation (in lattice)
+ bottom ("\<bottom>") "bottom == order_syntax.bottom carrier le"
+abbreviation (in total_order)
+ top ("\<top>") "top == order_syntax.top carrier le"
+abbreviation (in total_order)
+ bottom ("\<bottom>") "bottom == order_syntax.bottom carrier le"
+abbreviation (in complete_lattice)
+ top ("\<top>") "top == order_syntax.top carrier le"
+abbreviation (in complete_lattice)
+ bottom ("\<bottom>") "bottom == order_syntax.bottom carrier le"
lemma (in complete_lattice) supI:
- "[| !!l. least L l (Upper L A) ==> P l; A \<subseteq> carrier L |]
+ "[| !!l. least l (Upper A) ==> P l; A \<subseteq> carrier |]
==> P (\<Squnion>A)"
proof (unfold sup_def)
- assume L: "A \<subseteq> carrier L"
- and P: "!!l. least L l (Upper L A) ==> P l"
- with sup_exists obtain s where "least L s (Upper L A)" by blast
- with L show "P (THE l. least L l (Upper L A))"
+ assume L: "A \<subseteq> carrier"
+ and P: "!!l. least l (Upper A) ==> P l"
+ with sup_exists obtain s where "least s (Upper A)" by blast
+ with L show "P (THE l. least l (Upper A))"
by (fast intro: theI2 least_unique P)
qed
lemma (in complete_lattice) sup_closed [simp]:
- "A \<subseteq> carrier L ==> \<Squnion>A \<in> carrier L"
+ "A \<subseteq> carrier ==> \<Squnion>A \<in> carrier"
by (rule supI) simp_all
lemma (in complete_lattice) top_closed [simp, intro]:
- "\<top> \<in> carrier L"
+ "\<top> \<in> carrier"
by (unfold top_def) simp
lemma (in complete_lattice) infI:
- "[| !!i. greatest L i (Lower L A) ==> P i; A \<subseteq> carrier L |]
+ "[| !!i. greatest i (Lower A) ==> P i; A \<subseteq> carrier |]
==> P (\<Sqinter>A)"
proof (unfold inf_def)
- assume L: "A \<subseteq> carrier L"
- and P: "!!l. greatest L l (Lower L A) ==> P l"
- with inf_exists obtain s where "greatest L s (Lower L A)" by blast
- with L show "P (THE l. greatest L l (Lower L A))"
+ assume L: "A \<subseteq> carrier"
+ and P: "!!l. greatest l (Lower A) ==> P l"
+ with inf_exists obtain s where "greatest s (Lower A)" by blast
+ with L show "P (THE l. greatest l (Lower A))"
by (fast intro: theI2 greatest_unique P)
qed
lemma (in complete_lattice) inf_closed [simp]:
- "A \<subseteq> carrier L ==> \<Sqinter>A \<in> carrier L"
+ "A \<subseteq> carrier ==> \<Sqinter>A \<in> carrier"
by (rule infI) simp_all
lemma (in complete_lattice) bottom_closed [simp, intro]:
- "\<bottom> \<in> carrier L"
+ "\<bottom> \<in> carrier"
by (unfold bottom_def) simp
text {* Jacobson: Theorem 8.1 *}
-lemma Lower_empty [simp]:
- "Lower L {} = carrier L"
+lemma (in order_syntax) Lower_empty [simp]:
+ "Lower {} = carrier"
by (unfold Lower_def) simp
-lemma Upper_empty [simp]:
- "Upper L {} = carrier L"
+lemma (in order_syntax) Upper_empty [simp]:
+ "Upper {} = carrier"
by (unfold Upper_def) simp
theorem (in partial_order) complete_lattice_criterion1:
- assumes top_exists: "EX g. greatest L g (carrier L)"
+ assumes top_exists: "EX g. greatest g (carrier)"
and inf_exists:
- "!!A. [| A \<subseteq> carrier L; A ~= {} |] ==> EX i. greatest L i (Lower L A)"
- shows "complete_lattice L"
+ "!!A. [| A \<subseteq> carrier; A ~= {} |] ==> EX i. greatest i (Lower A)"
+ shows "complete_lattice carrier le"
proof (rule complete_latticeI)
- from top_exists obtain top where top: "greatest L top (carrier L)" ..
+ from top_exists obtain top where top: "greatest top (carrier)" ..
fix A
- assume L: "A \<subseteq> carrier L"
- let ?B = "Upper L A"
+ assume L: "A \<subseteq> carrier"
+ let ?B = "Upper A"
from L top have "top \<in> ?B" by (fast intro!: Upper_memI intro: greatest_le)
then have B_non_empty: "?B ~= {}" by fast
- have B_L: "?B \<subseteq> carrier L" by simp
+ have B_L: "?B \<subseteq> carrier" by simp
from inf_exists [OF B_L B_non_empty]
- obtain b where b_inf_B: "greatest L b (Lower L ?B)" ..
- have "least L b (Upper L A)"
+ obtain b where b_inf_B: "greatest b (Lower ?B)" ..
+ have "least b (Upper A)"
apply (rule least_UpperI)
- apply (rule greatest_le [where A = "Lower L ?B"])
+ apply (rule greatest_le [where A = "Lower ?B"])
apply (rule b_inf_B)
apply (rule Lower_memI)
apply (erule UpperD)
@@ -808,11 +913,11 @@
apply (rule L)
apply (rule greatest_carrier [OF b_inf_B]) (* rename rule: _closed *)
done
- then show "EX s. least L s (Upper L A)" ..
+ then show "EX s. least s (Upper A)" ..
next
fix A
- assume L: "A \<subseteq> carrier L"
- show "EX i. greatest L i (Lower L A)"
+ assume L: "A \<subseteq> carrier"
+ show "EX i. greatest i (Lower A)"
proof (cases "A = {}")
case True then show ?thesis
by (simp add: top_exists)
@@ -830,25 +935,25 @@
subsubsection {* Powerset of a Set is a Complete Lattice *}
theorem powerset_is_complete_lattice:
- "complete_lattice (| carrier = Pow A, le = op \<subseteq> |)"
- (is "complete_lattice ?L")
+ "complete_lattice (Pow A) (op \<subseteq>)"
+ (is "complete_lattice ?car ?le")
proof (rule partial_order.complete_latticeI)
- show "partial_order ?L"
+ show "partial_order ?car ?le"
by (rule partial_order.intro) auto
next
fix B
- assume "B \<subseteq> carrier ?L"
- then have "least ?L (\<Union> B) (Upper ?L B)"
- by (fastsimp intro!: least_UpperI simp: Upper_def)
- then show "EX s. least ?L s (Upper ?L B)" ..
+ assume "B \<subseteq> ?car"
+ then have "order_syntax.least ?car ?le (\<Union> B) (order_syntax.Upper ?car ?le B)"
+ by (fastsimp intro!: order_syntax.least_UpperI simp: order_syntax.Upper_def)
+ then show "EX s. order_syntax.least ?car ?le s (order_syntax.Upper ?car ?le B)" ..
next
fix B
- assume "B \<subseteq> carrier ?L"
- then have "greatest ?L (\<Inter> B \<inter> A) (Lower ?L B)"
+ assume "B \<subseteq> ?car"
+ then have "order_syntax.greatest ?car ?le (\<Inter> B \<inter> A) (order_syntax.Lower ?car ?le B)"
txt {* @{term "\<Inter> B"} is not the infimum of @{term B}:
@{term "\<Inter> {} = UNIV"} which is in general bigger than @{term "A"}! *}
- by (fastsimp intro!: greatest_LowerI simp: Lower_def)
- then show "EX i. greatest ?L i (Lower ?L B)" ..
+ by (fastsimp intro!: order_syntax.greatest_LowerI simp: order_syntax.Lower_def)
+ then show "EX i. order_syntax.greatest ?car ?le i (order_syntax.Lower ?car ?le B)" ..
qed
text {* An other example, that of the lattice of subgroups of a group,