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author | ballarin |

Tue, 13 Apr 2004 09:42:40 +0200 | |

changeset 14551 | 2cb6ff394bfb |

parent 14550 | b13da5649bf9 |

child 14552 | e88f52b775a5 |

Various changes to HOL-Algebra;
Locale instantiation.

--- a/NEWS Tue Apr 13 07:48:32 2004 +0200 +++ b/NEWS Tue Apr 13 09:42:40 2004 +0200 @@ -85,13 +85,17 @@ - Rule sets <locale>.intro and <locale>.axioms no longer declared as [intro?] and [elim?] (respectively) by default. - Experimental command for instantiation of locales in proof contexts: - instantiate <label>: <loc> + instantiate <label>[<attrs>]: <loc> Instantiates locale <loc> and adds all its theorems to the current context - taking into account their attributes, and qualifying their names with - <label>. If the locale has assumptions, a chained fact of the form + taking into account their attributes. Label and attrs are optional + modifiers, like in theorem declarations. If present, names of + instantiated theorems are qualified with <label>, and the attributes + <attrs> are applied after any attributes these theorems might have already. + If the locale has assumptions, a chained fact of the form "<loc> t1 ... tn" is expected from which instantiations of the parameters - are derived. - A few (very simple) examples can be found in FOL/ex/LocaleInst.thy. + are derived. The command does not support old-style locales declared + with "locale (open)". + A few (very simple) examples can be found in FOL/ex/LocaleInst.thy. * HOL: Tactic emulation methods induct_tac and case_tac understand static (Isar) contexts.

--- a/src/FOL/ex/LocaleInst.thy Tue Apr 13 07:48:32 2004 +0200 +++ b/src/FOL/ex/LocaleInst.thy Tue Apr 13 09:42:40 2004 +0200 @@ -18,10 +18,18 @@ lemma "[| A; B |] ==> A & B" proof - - instantiate my: L1 txt {* No chained fact required. *} - assume B and A txt {* order reversed *} + instantiate my: L1 txt {* No chained fact required. *} + assume B and A txt {* order reversed *} + then show "A & B" .. txt {* Applies @{thm my.rev_conjI}. *} +qed + +locale L11 = notes rev_conjI = conjI [THEN iffD1 [OF conj_commute]] + +lemma "[| A; B |] ==> A & B" +proof - + instantiate [intro]: L11 txt {* Attribute supplied at instantiation. *} + assume B and A then show "A & B" .. - txt {* Applies @{thm my.rev_conjI}. *} qed section {* Simple locale with assumptions *} @@ -111,4 +119,15 @@ show ?thesis by (rule lem) (* lem instantiated to True *) qed +section {* Instantiation in a context with target *} + +lemma (in L4) (* Target might confuse instantiation command. *) + fixes A (infixl "$" 60) + assumes A: "L4(A)" + shows "(x::i) $ y $ z $ w = y $ x $ w $ z" +proof - + from A instantiate A: L4 + show ?thesis by (simp only: A.OP.AC) +qed + end

--- a/src/HOL/Algebra/CRing.thy Tue Apr 13 07:48:32 2004 +0200 +++ b/src/HOL/Algebra/CRing.thy Tue Apr 13 09:42:40 2004 +0200 @@ -302,6 +302,9 @@ and integral: "[| a \<otimes> b = \<zero>; a \<in> carrier R; b \<in> carrier R |] ==> a = \<zero> | b = \<zero>" +locale field = "domain" + + assumes field_Units: "Units R = carrier R - {\<zero>}" + subsection {* Basic Facts of Rings *} lemma ringI: @@ -357,7 +360,7 @@ "comm_monoid R" by (auto intro!: comm_monoidI m_assoc m_comm) -subsection {* Normaliser for Commutative Rings *} +subsection {* Normaliser for Rings *} lemma (in abelian_group) r_neg2: "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<oplus> (\<ominus> x \<oplus> y) = y"

--- a/src/HOL/Algebra/Group.thy Tue Apr 13 07:48:32 2004 +0200 +++ b/src/HOL/Algebra/Group.thy Tue Apr 13 09:42:40 2004 +0200 @@ -410,6 +410,7 @@ shows "semigroup (G(| carrier := H |))" using prems by fast + locale subgroup = submagma H G + assumes one_closed [intro, simp]: "\<one> \<in> H" and m_inv_closed [intro, simp]: "x \<in> H ==> inv x \<in> H"

--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Algebra/Lattice.thy Tue Apr 13 09:42:40 2004 +0200 @@ -0,0 +1,985 @@ +(* + Title: Orders and Lattices + Id: $Id$ + Author: Clemens Ballarin, started 7 November 2003 + Copyright: Clemens Ballarin +*) + +theory Lattice = Group: + +section {* Order and Lattices *} + +subsection {* Partial Orders *} + +record 'a order = "'a partial_object" + + le :: "['a, 'a] => bool" (infixl "\<sqsubseteq>\<index>" 50) + +locale order_syntax = struct L + +locale partial_order = order_syntax + + assumes refl [intro, simp]: + "x \<in> carrier L ==> x \<sqsubseteq> x" + and anti_sym [intro]: + "[| x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> carrier L; y \<in> carrier L |] ==> 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 + less :: "[('a, 'm) order_scheme, 'a, 'a] => bool" (infixl "\<sqsubset>\<index>" 50) + "less L x y == le L x y & x ~= y" + + (* Upper and lower bounds of a set. *) + Upper :: "[('a, 'm) order_scheme, 'a set] => 'a set" + "Upper L A == {u. (ALL x. x \<in> A \<inter> carrier L --> le L x u)} \<inter> + carrier L" + + Lower :: "[('a, 'm) order_scheme, 'a set] => 'a set" + "Lower L A == {l. (ALL x. x \<in> A \<inter> carrier L --> le L l x)} \<inter> + carrier L" + + (* Least and greatest, as predicate. *) + least :: "[('a, 'm) order_scheme, 'a, 'a set] => bool" + "least L l A == A \<subseteq> carrier L & l \<in> A & (ALL x : A. le L l x)" + + greatest :: "[('a, 'm) order_scheme, 'a, 'a set] => bool" + "greatest L g A == A \<subseteq> carrier L & g \<in> A & (ALL x : A. le L x g)" + + (* Supremum and infimum *) + sup :: "[('a, 'm) order_scheme, 'a set] => 'a" ("\<Squnion>\<index>_" [90] 90) + "sup L A == THE x. least L x (Upper L A)" + + inf :: "[('a, 'm) order_scheme, 'a set] => 'a" ("\<Sqinter>\<index>_" [90] 90) + "inf L A == THE x. greatest L x (Lower L A)" + + join :: "[('a, 'm) order_scheme, 'a, 'a] => 'a" (infixl "\<squnion>\<index>" 65) + "join L x y == sup L {x, y}" + + meet :: "[('a, 'm) order_scheme, 'a, 'a] => 'a" (infixl "\<sqinter>\<index>" 65) + "meet L x y == inf L {x, y}" + +(* Upper *) + +lemma Upper_closed [intro, simp]: + "Upper L A \<subseteq> carrier L" + by (unfold Upper_def) clarify + +lemma UpperD [dest]: + includes order_syntax + shows "[| u \<in> Upper L A; x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> u" + by (unfold Upper_def) blast + +lemma Upper_memI: + includes order_syntax + shows "[| !! y. y \<in> A ==> y \<sqsubseteq> x; x \<in> carrier L |] ==> x \<in> Upper L A" + by (unfold Upper_def) blast + +lemma Upper_antimono: + "A \<subseteq> B ==> Upper L B \<subseteq> Upper L A" + by (unfold Upper_def) blast + +(* Lower *) + +lemma Lower_closed [intro, simp]: + "Lower L A \<subseteq> carrier L" + by (unfold Lower_def) clarify + +lemma LowerD [dest]: + includes order_syntax + shows "[| l \<in> Lower L A; x \<in> A; A \<subseteq> carrier L |] ==> l \<sqsubseteq> x" + by (unfold Lower_def) blast + +lemma Lower_memI: + includes order_syntax + shows "[| !! y. y \<in> A ==> x \<sqsubseteq> y; x \<in> carrier L |] ==> x \<in> Lower L A" + by (unfold Lower_def) blast + +lemma Lower_antimono: + "A \<subseteq> B ==> Lower L B \<subseteq> Lower L A" + by (unfold Lower_def) blast + +(* least *) + +lemma least_carrier [intro, simp]: + shows "least L l A ==> l \<in> carrier L" + by (unfold least_def) fast + +lemma least_mem: + "least L 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" + by (unfold least_def) blast + +lemma least_le: + includes order_syntax + shows "[| least L x A; a \<in> A |] ==> x \<sqsubseteq> a" + by (unfold least_def) fast + +lemma least_UpperI: + includes order_syntax + 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)" +proof (unfold least_def, intro conjI) + show "Upper L A \<subseteq> carrier L" by simp +next + from above L show "s \<in> Upper L A" by (simp add: Upper_def) +next + from below show "ALL x : Upper L A. s \<sqsubseteq> x" by fast +qed + +(* greatest *) + +lemma greatest_carrier [intro, simp]: + shows "greatest L l A ==> l \<in> carrier L" + by (unfold greatest_def) fast + +lemma greatest_mem: + "greatest L 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" + by (unfold greatest_def) blast + +lemma greatest_le: + includes order_syntax + shows "[| greatest L x A; a \<in> A |] ==> a \<sqsubseteq> x" + by (unfold greatest_def) fast + +lemma greatest_LowerI: + includes order_syntax + 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)" +proof (unfold greatest_def, intro conjI) + show "Lower L A \<subseteq> carrier L" by simp +next + from below L show "i \<in> Lower L A" by (simp add: Lower_def) +next + from above show "ALL x : Lower L A. x \<sqsubseteq> i" by fast +qed + +subsection {* Lattices *} + +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})" + and inf_of_two_exists: + "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. greatest L s (Lower L {x, y})" + +lemma least_Upper_above: + includes order_syntax + shows "[| least L s (Upper L A); x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> s" + by (unfold least_def) blast + +lemma greatest_Lower_above: + includes order_syntax + shows "[| greatest L i (Lower L A); x \<in> A; A \<subseteq> carrier L |] ==> 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 |] + ==> 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}))" + 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" + 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})" + by (rule least_UpperI) fast+ + +lemma (in partial_order) sup_of_singleton [simp]: + includes order_syntax + shows "x \<in> carrier L ==> \<Squnion> {x} = x" + by (unfold sup_def) (blast intro: least_unique least_UpperI sup_of_singletonI) + +text {* Condition on 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 |] + ==> 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 + 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)))" + proof (rule theI2 [where a = s]) + show "least L s (Upper L (insert x A))" + proof (rule least_UpperI) + fix z + assume xA: "z \<in> insert x A" + show "z \<sqsubseteq> s" + proof - + { + assume "z = x" then have ?thesis + by (simp add: least_Upper_above [OF least_s] L La) + } + moreover + { + assume "z \<in> A" + with L least_s least_a have ?thesis + by (rule_tac trans [where y = a]) (auto dest: least_Upper_above) + } + moreover note xA + ultimately show ?thesis by blast + qed + next + fix y + assume y: "y \<in> Upper L (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}" + show "z \<sqsubseteq> y" + proof - + { + have y': "y \<in> Upper L A" + apply (rule subsetD [where A = "Upper L (insert x A)"]) + apply (rule Upper_antimono) apply clarify apply assumption + done + assume "z = a" + with y' least_a have ?thesis by (fast dest: least_le) + } + moreover + { + assume "z = x" + with y L have ?thesis by blast + } + moreover note z + ultimately show ?thesis by blast + qed + qed (rule Upper_closed [THEN subsetD]) + next + from L show "insert x A \<subseteq> carrier L" by simp + next + from least_s show "s \<in> carrier L" by simp + qed +next + fix l + assume least_l: "least L l (Upper L (insert x A))" + show "l = s" + proof (rule least_unique) + show "least L s (Upper L (insert x A))" + proof (rule least_UpperI) + fix z + assume xA: "z \<in> insert x A" + show "z \<sqsubseteq> s" + proof - + { + assume "z = x" then have ?thesis + by (simp add: least_Upper_above [OF least_s] L La) + } + moreover + { + assume "z \<in> A" + with L least_s least_a have ?thesis + by (rule_tac trans [where y = a]) (auto dest: least_Upper_above) + } + moreover note xA + ultimately show ?thesis by blast + qed + next + fix y + assume y: "y \<in> Upper L (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}" + show "z \<sqsubseteq> y" + proof - + { + have y': "y \<in> Upper L A" + apply (rule subsetD [where A = "Upper L (insert x A)"]) + apply (rule Upper_antimono) apply clarify apply assumption + done + assume "z = a" + with y' least_a have ?thesis by (fast dest: least_le) + } + moreover + { + assume "z = x" + with y L have ?thesis by blast + } + moreover note z + ultimately show ?thesis by blast + qed + qed (rule Upper_closed [THEN subsetD]) + next + from L show "insert x A \<subseteq> carrier L" by simp + next + from least_s show "s \<in> carrier L" 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)" +proof (induct set: Finites) + case empty then show ?case by simp +next + case (insert A x) + show ?case + proof (cases "A = {}") + case True + with insert show ?thesis by (simp add: sup_of_singletonI) + next + case False + from insert show ?thesis + proof (rule_tac sup_insertI) + from False insert show "least L (\<Squnion> A) (Upper L A)" by simp + qed simp_all + 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" + shows "P (\<Squnion> (insert x A))" +proof (cases "A = {}") + case True with P and xA show ?thesis + by (simp add: sup_of_singletonI) +next + case False with P and xA show ?thesis + by (simp add: sup_insertI finite_sup_least) +qed + +lemma (in lattice) finite_sup_closed: + "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Squnion> A \<in> carrier L" +proof (induct set: Finites) + case empty then show ?case by simp +next + case (insert A x) then show ?case + by (rule_tac finite_sup_insertI) (simp_all) +qed + +lemma (in lattice) join_left: + "[| x \<in> carrier L; y \<in> carrier L |] ==> 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" + 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})" +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})" + 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" + shows "x \<squnion> y \<sqsubseteq> z" +proof (rule joinI) + fix s + assume "least L s (Upper L {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" + 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})" + show "x \<squnion> (y \<squnion> z) = s" + proof (rule anti_sym) + from sup L show "x \<squnion> (y \<squnion> z) \<sqsubseteq> s" + by (fastsimp intro!: join_le elim: least_Upper_above) + next + from sup L show "s \<sqsubseteq> x \<squnion> (y \<squnion> z)" + by (erule_tac least_le) + (blast intro!: Upper_memI intro: trans join_left join_right join_closed) + qed (simp_all add: L least_carrier [OF sup]) +qed (simp_all add: L) + +lemma join_comm: + includes order_syntax + shows "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" + 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) + also from L have "... = \<Squnion> {z, x, y}" by (simp add: join_assoc_lemma) + also from L have "... = \<Squnion> {x, y, z}" by (simp add: insert_commute) + also from L have "... = x \<squnion> (y \<squnion> z)" by (simp add: join_assoc_lemma) + finally show ?thesis . +qed + +subsubsection {* Infimum *} + +lemma (in lattice) meetI: + "[| !!i. greatest L i (Lower L {x, y}) ==> P i; + x \<in> carrier L; y \<in> carrier L |] + ==> 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}))" + 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" + 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})" + by (rule greatest_LowerI) fast+ + +lemma (in partial_order) inf_of_singleton [simp]: + includes order_syntax + shows "x \<in> carrier L ==> \<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 |] + ==> 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 + 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)))" + proof (rule theI2 [where a = i]) + show "greatest L i (Lower L (insert x A))" + proof (rule greatest_LowerI) + fix z + assume xA: "z \<in> insert x A" + show "i \<sqsubseteq> z" + proof - + { + assume "z = x" then have ?thesis + by (simp add: greatest_Lower_above [OF greatest_i] L La) + } + moreover + { + assume "z \<in> A" + with L greatest_i greatest_a have ?thesis + by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above) + } + moreover note xA + ultimately show ?thesis by blast + qed + next + fix y + assume y: "y \<in> Lower L (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}" + show "y \<sqsubseteq> z" + proof - + { + have y': "y \<in> Lower L A" + apply (rule subsetD [where A = "Lower L (insert x A)"]) + apply (rule Lower_antimono) apply clarify apply assumption + done + assume "z = a" + with y' greatest_a have ?thesis by (fast dest: greatest_le) + } + moreover + { + assume "z = x" + with y L have ?thesis by blast + } + moreover note z + ultimately show ?thesis by blast + qed + qed (rule Lower_closed [THEN subsetD]) + next + from L show "insert x A \<subseteq> carrier L" by simp + next + from greatest_i show "i \<in> carrier L" by simp + qed +next + fix g + assume greatest_g: "greatest L g (Lower L (insert x A))" + show "g = i" + proof (rule greatest_unique) + show "greatest L i (Lower L (insert x A))" + proof (rule greatest_LowerI) + fix z + assume xA: "z \<in> insert x A" + show "i \<sqsubseteq> z" + proof - + { + assume "z = x" then have ?thesis + by (simp add: greatest_Lower_above [OF greatest_i] L La) + } + moreover + { + assume "z \<in> A" + with L greatest_i greatest_a have ?thesis + by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above) + } + moreover note xA + ultimately show ?thesis by blast + qed + next + fix y + assume y: "y \<in> Lower L (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}" + show "y \<sqsubseteq> z" + proof - + { + have y': "y \<in> Lower L A" + apply (rule subsetD [where A = "Lower L (insert x A)"]) + apply (rule Lower_antimono) apply clarify apply assumption + done + assume "z = a" + with y' greatest_a have ?thesis by (fast dest: greatest_le) + } + moreover + { + assume "z = x" + with y L have ?thesis by blast + } + moreover note z + ultimately show ?thesis by blast + qed + qed (rule Lower_closed [THEN subsetD]) + next + from L show "insert x A \<subseteq> carrier L" by simp + next + from greatest_i show "i \<in> carrier L" 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)" +proof (induct set: Finites) + case empty then show ?case by simp +next + case (insert A x) + show ?case + proof (cases "A = {}") + case True + with insert show ?thesis by (simp add: inf_of_singletonI) + next + case False + from insert show ?thesis + proof (rule_tac inf_insertI) + from False insert show "greatest L (\<Sqinter> A) (Lower L 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" + shows "P (\<Sqinter> (insert x A))" +proof (cases "A = {}") + case True with P and xA show ?thesis + by (simp add: inf_of_singletonI) +next + case False with P and xA show ?thesis + by (simp add: inf_insertI finite_inf_greatest) +qed + +lemma (in lattice) finite_inf_closed: + "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Sqinter> A \<in> carrier L" +proof (induct set: Finites) + case empty then show ?case by simp +next + case (insert A x) then show ?case + by (rule_tac finite_inf_insertI) (simp_all) +qed + +lemma (in lattice) meet_left: + "[| x \<in> carrier L; y \<in> carrier L |] ==> 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" + 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})" +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 + with L + show "greatest L (THE xa. greatest L xa (Lower L {x, y})) (Lower L {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" + shows "z \<sqsubseteq> x \<sqinter> y" +proof (rule meetI) + fix i + assume "greatest L i (Lower L {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" + 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})" + show "x \<sqinter> (y \<sqinter> z) = i" + proof (rule anti_sym) + from inf L show "i \<sqsubseteq> x \<sqinter> (y \<sqinter> z)" + by (fastsimp intro!: meet_le elim: greatest_Lower_above) + next + from inf L show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> i" + by (erule_tac greatest_le) + (blast intro!: Lower_memI intro: trans meet_left meet_right meet_closed) + qed (simp_all add: L greatest_carrier [OF inf]) +qed (simp_all add: L) + +lemma meet_comm: + includes order_syntax + shows "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" + 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) + also from L have "... = \<Sqinter> {z, x, y}" by (simp add: meet_assoc_lemma) + also from L have "... = \<Sqinter> {x, y, z}" by (simp add: insert_commute) + also from L have "... = x \<sqinter> (y \<sqinter> z)" by (simp add: meet_assoc_lemma) + finally show ?thesis . +qed + +subsection {* Total Orders *} + +locale total_order = lattice + + assumes total: "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> y | y \<sqsubseteq> x" + +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" +proof (rule total_order.intro) + show "lattice_axioms L" + 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})" + proof - + note total L + moreover + { + assume "x \<sqsubseteq> y" + with L have "least L y (Upper L {x, y})" + by (rule_tac least_UpperI) auto + } + moreover + { + assume "y \<sqsubseteq> x" + with L have "least L x (Upper L {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})" + proof - + note total L + moreover + { + assume "y \<sqsubseteq> x" + with L have "greatest L y (Lower L {x, y})" + by (rule_tac greatest_LowerI) auto + } + moreover + { + assume "x \<sqsubseteq> y" + with L have "greatest L x (Lower L {x, y})" + by (rule_tac greatest_LowerI) auto + } + ultimately show ?thesis by blast + qed + qed +qed (assumption | rule total_order_axioms.intro)+ + +subsection {* Complete lattices *} + +locale complete_lattice = lattice + + assumes sup_exists: + "[| A \<subseteq> carrier L |] ==> EX s. least L s (Upper L A)" + and inf_exists: + "[| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)" + +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)" + and inf_exists: + "!!A. [| A \<subseteq> carrier L |] ==> EX i. greatest L i (Lower L A)" + shows "complete_lattice L" +proof (rule complete_lattice.intro) + show "lattice_axioms L" + by (rule lattice_axioms.intro) (blast intro: sup_exists inf_exists)+ +qed (assumption | rule complete_lattice_axioms.intro)+ + +constdefs + top :: "('a, 'm) order_scheme => 'a" ("\<top>\<index>") + "top L == sup L (carrier L)" + + bottom :: "('a, 'm) order_scheme => 'a" ("\<bottom>\<index>") + "bottom L == inf L (carrier L)" + + +lemma (in complete_lattice) supI: + "[| !!l. least L l (Upper L A) ==> P l; A \<subseteq> carrier L |] + ==> 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))" + by (fast intro: theI2 least_unique P) +qed + +lemma (in complete_lattice) sup_closed [simp]: + "A \<subseteq> carrier L ==> \<Squnion> A \<in> carrier L" + by (rule supI) simp_all + +lemma (in complete_lattice) top_closed [simp, intro]: + "\<top> \<in> carrier L" + by (unfold top_def) simp + +lemma (in complete_lattice) infI: + "[| !!i. greatest L i (Lower L A) ==> P i; A \<subseteq> carrier L |] + ==> 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))" + by (fast intro: theI2 greatest_unique P) +qed + +lemma (in complete_lattice) inf_closed [simp]: + "A \<subseteq> carrier L ==> \<Sqinter> A \<in> carrier L" + by (rule infI) simp_all + +lemma (in complete_lattice) bottom_closed [simp, intro]: + "\<bottom> \<in> carrier L" + by (unfold bottom_def) simp + +text {* Jacobson: Theorem 8.1 *} + +lemma Lower_empty [simp]: + "Lower L {} = carrier L" + by (unfold Lower_def) simp + +lemma Upper_empty [simp]: + "Upper L {} = carrier L" + by (unfold Upper_def) simp + +theorem (in partial_order) complete_lattice_criterion1: + assumes top_exists: "EX g. greatest L g (carrier L)" + and inf_exists: + "!!A. [| A \<subseteq> carrier L; A ~= {} |] ==> EX i. greatest L i (Lower L A)" + shows "complete_lattice L" +proof (rule complete_latticeI) + from top_exists obtain top where top: "greatest L top (carrier L)" .. + fix A + assume L: "A \<subseteq> carrier L" + let ?B = "Upper L 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 + 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)" +apply (rule least_UpperI) + apply (rule greatest_le [where A = "Lower L ?B"]) + apply (rule b_inf_B) + apply (rule Lower_memI) + apply (erule UpperD) + apply assumption + apply (rule L) + apply (fast intro: L [THEN subsetD]) + apply (erule greatest_Lower_above [OF b_inf_B]) + apply simp + 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)" .. +next + fix A + assume L: "A \<subseteq> carrier L" + show "EX i. greatest L i (Lower L A)" + proof (cases "A = {}") + case True then show ?thesis + by (simp add: top_exists) + next + case False with L show ?thesis + by (rule inf_exists) + qed +qed + +(* TODO: prove dual version *) + +subsection {* Examples *} + +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") +proof (rule partial_order.complete_latticeI) + show "partial_order ?L" + 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)" .. +next + fix B + assume "B \<subseteq> carrier ?L" + then have "greatest ?L (\<Inter> B \<inter> A) (Lower ?L 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)" .. +qed + +subsubsection {* Lattice of subgroups of a group *} + +theorem (in group) subgroups_partial_order: + "partial_order (| carrier = {H. subgroup H G}, le = op \<subseteq> |)" + by (rule partial_order.intro) simp_all + +lemma (in group) subgroup_self: + "subgroup (carrier G) G" + by (rule subgroupI) auto + +lemma (in group) subgroup_imp_group: + "subgroup H G ==> group (G(| carrier := H |))" + using subgroup.groupI [OF _ group.intro] . + +lemma (in group) is_monoid [intro, simp]: + "monoid G" + by (rule monoid.intro) + +lemma (in group) subgroup_inv_equality: + "[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x" +apply (rule_tac inv_equality [THEN sym]) + apply (rule group.l_inv [OF subgroup_imp_group, simplified]) + apply assumption+ + apply (rule subsetD [OF subgroup.subset]) + apply assumption+ +apply (rule subsetD [OF subgroup.subset]) + apply assumption +apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified]) + apply assumption+ +done + +theorem (in group) subgroups_Inter: + assumes subgr: "(!!H. H \<in> A ==> subgroup H G)" + and not_empty: "A ~= {}" + shows "subgroup (\<Inter>A) G" +proof (rule subgroupI) + from subgr [THEN subgroup.subset] and not_empty + show "\<Inter>A \<subseteq> carrier G" by blast +next + from subgr [THEN subgroup.one_closed] + show "\<Inter>A ~= {}" by blast +next + fix x assume "x \<in> \<Inter>A" + with subgr [THEN subgroup.m_inv_closed] + show "inv x \<in> \<Inter>A" by blast +next + fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A" + with subgr [THEN subgroup.m_closed] + show "x \<otimes> y \<in> \<Inter>A" by blast +qed + +theorem (in group) subgroups_complete_lattice: + "complete_lattice (| carrier = {H. subgroup H G}, le = op \<subseteq> |)" + (is "complete_lattice ?L") +proof (rule partial_order.complete_lattice_criterion1) + show "partial_order ?L" by (rule subgroups_partial_order) +next + have "greatest ?L (carrier G) (carrier ?L)" + by (unfold greatest_def) (simp add: subgroup.subset subgroup_self) + then show "EX G. greatest ?L G (carrier ?L)" .. +next + fix A + assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}" + then have Int_subgroup: "subgroup (\<Inter>A) G" + by (fastsimp intro: subgroups_Inter) + have "greatest ?L (\<Inter>A) (Lower ?L A)" + (is "greatest ?L ?Int _") + proof (rule greatest_LowerI) + fix H + assume H: "H \<in> A" + with L have subgroupH: "subgroup H G" by auto + from subgroupH have submagmaH: "submagma H G" by (rule subgroup.axioms) + from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H") + by (rule subgroup_imp_group) + from groupH have monoidH: "monoid ?H" + by (rule group.is_monoid) + from H have Int_subset: "?Int \<subseteq> H" by fastsimp + then show "le ?L ?Int H" by simp + next + fix H + assume H: "H \<in> Lower ?L A" + with L Int_subgroup show "le ?L H ?Int" by (fastsimp intro: Inter_greatest) + next + show "A \<subseteq> carrier ?L" by (rule L) + next + show "?Int \<in> carrier ?L" by simp (rule Int_subgroup) + qed + then show "EX I. greatest ?L I (Lower ?L A)" .. +qed + +end \ No newline at end of file

--- a/src/HOL/Algebra/ROOT.ML Tue Apr 13 07:48:32 2004 +0200 +++ b/src/HOL/Algebra/ROOT.ML Tue Apr 13 09:42:40 2004 +0200 @@ -16,6 +16,7 @@ use_thy "FiniteProduct"; (* Product operator for commutative groups *) use_thy "Sylow"; (* Sylow's theorem *) use_thy "Bij"; (* Automorphism Groups *) +use_thy "Lattice"; (* Lattices, and the complete lattice of subgroups *) (* Rings *)

--- a/src/HOL/Algebra/document/root.tex Tue Apr 13 07:48:32 2004 +0200 +++ b/src/HOL/Algebra/document/root.tex Tue Apr 13 09:42:40 2004 +0200 @@ -22,7 +22,7 @@ % \<twosuperior>, \<onehalf>, % \<threesuperior>, \<threequarters> % \<degree> -%\usepackage[only,bigsqcap]{stmaryrd} % for \<Sqinter> +\usepackage[only,bigsqcap]{stmaryrd} % for \<Sqinter> %\usepackage{wasysym} %\usepackage{eufrak} % for \<AA> ... \<ZZ>, \<aa> ... \<zz> %\usepackage{textcomp} % for \<zero> ... \<nine>, \<cent>

--- a/src/HOL/IsaMakefile Tue Apr 13 07:48:32 2004 +0200 +++ b/src/HOL/IsaMakefile Tue Apr 13 09:42:40 2004 +0200 @@ -341,6 +341,7 @@ Algebra/Exponent.thy \ Algebra/FiniteProduct.thy \ Algebra/Group.thy \ + Algebra/Lattice.thy \ Algebra/Module.thy \ Algebra/Sylow.thy \ Algebra/UnivPoly.thy \

--- a/src/HOL/Set.thy Tue Apr 13 07:48:32 2004 +0200 +++ b/src/HOL/Set.thy Tue Apr 13 09:42:40 2004 +0200 @@ -943,6 +943,10 @@ lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B" by blast +lemma Inter_subset: + "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B" + by blast + lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A" by (rules intro: InterI subsetI dest: subsetD)