author | ballarin |
Wed, 17 Dec 2008 17:53:56 +0100 | |
changeset 29240 | bb81c3709fb6 |
parent 29237 | e90d9d51106b |
child 30729 | 461ee3e49ad3 |
permissions | -rw-r--r-- |
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(* |
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Title: HOL/Algebra/Group.thy |
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Author: Clemens Ballarin, started 4 February 2003 |
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Based on work by Florian Kammueller, L C Paulson and Markus Wenzel. |
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*) |
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theory Group |
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imports Lattice FuncSet |
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begin |
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section {* Monoids and Groups *} |
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Restructured algebra library, added ideals and quotient rings.
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subsection {* Definitions *} |
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Restructured algebra library, added ideals and quotient rings.
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|
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text {* |
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Definitions follow \cite{Jacobson:1985}. |
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*} |
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record 'a monoid = "'a partial_object" + |
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mult :: "['a, 'a] \<Rightarrow> 'a" (infixl "\<otimes>\<index>" 70) |
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one :: 'a ("\<one>\<index>") |
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constdefs (structure G) |
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m_inv :: "('a, 'b) monoid_scheme => 'a => 'a" ("inv\<index> _" [81] 80) |
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"inv x == (THE y. y \<in> carrier G & x \<otimes> y = \<one> & y \<otimes> x = \<one>)" |
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Units :: "_ => 'a set" |
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--{*The set of invertible elements*} |
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"Units G == {y. y \<in> carrier G & (\<exists>x \<in> carrier G. x \<otimes> y = \<one> & y \<otimes> x = \<one>)}" |
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consts |
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pow :: "[('a, 'm) monoid_scheme, 'a, 'b::number] => 'a" (infixr "'(^')\<index>" 75) |
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defs (overloaded) |
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nat_pow_def: "pow G a n == nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) n" |
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int_pow_def: "pow G a z == |
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let p = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) |
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in if neg z then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z)" |
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locale monoid = |
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fixes G (structure) |
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assumes m_closed [intro, simp]: |
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"\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> carrier G" |
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and m_assoc: |
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"\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> |
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\<Longrightarrow> (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
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and one_closed [intro, simp]: "\<one> \<in> carrier G" |
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and l_one [simp]: "x \<in> carrier G \<Longrightarrow> \<one> \<otimes> x = x" |
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and r_one [simp]: "x \<in> carrier G \<Longrightarrow> x \<otimes> \<one> = x" |
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lemma monoidI: |
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fixes G (structure) |
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assumes m_closed: |
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"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G" |
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and one_closed: "\<one> \<in> carrier G" |
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and m_assoc: |
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"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
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(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
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and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" |
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and r_one: "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x" |
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shows "monoid G" |
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by (fast intro!: monoid.intro intro: assms) |
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lemma (in monoid) Units_closed [dest]: |
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"x \<in> Units G ==> x \<in> carrier G" |
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by (unfold Units_def) fast |
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lemma (in monoid) inv_unique: |
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assumes eq: "y \<otimes> x = \<one>" "x \<otimes> y' = \<one>" |
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and G: "x \<in> carrier G" "y \<in> carrier G" "y' \<in> carrier G" |
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shows "y = y'" |
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proof - |
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from G eq have "y = y \<otimes> (x \<otimes> y')" by simp |
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also from G have "... = (y \<otimes> x) \<otimes> y'" by (simp add: m_assoc) |
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also from G eq have "... = y'" by simp |
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finally show ?thesis . |
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qed |
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lemma (in monoid) Units_m_closed [intro, simp]: |
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assumes x: "x \<in> Units G" and y: "y \<in> Units G" |
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shows "x \<otimes> y \<in> Units G" |
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proof - |
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from x obtain x' where x: "x \<in> carrier G" "x' \<in> carrier G" and xinv: "x \<otimes> x' = \<one>" "x' \<otimes> x = \<one>" |
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unfolding Units_def by fast |
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from y obtain y' where y: "y \<in> carrier G" "y' \<in> carrier G" and yinv: "y \<otimes> y' = \<one>" "y' \<otimes> y = \<one>" |
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unfolding Units_def by fast |
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from x y xinv yinv have "y' \<otimes> (x' \<otimes> x) \<otimes> y = \<one>" by simp |
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moreover from x y xinv yinv have "x \<otimes> (y \<otimes> y') \<otimes> x' = \<one>" by simp |
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moreover note x y |
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ultimately show ?thesis unfolding Units_def |
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-- "Must avoid premature use of @{text hyp_subst_tac}." |
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apply (rule_tac CollectI) |
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apply (rule) |
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apply (fast) |
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apply (rule bexI [where x = "y' \<otimes> x'"]) |
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apply (auto simp: m_assoc) |
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done |
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qed |
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lemma (in monoid) Units_one_closed [intro, simp]: |
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"\<one> \<in> Units G" |
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by (unfold Units_def) auto |
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lemma (in monoid) Units_inv_closed [intro, simp]: |
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"x \<in> Units G ==> inv x \<in> carrier G" |
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apply (unfold Units_def m_inv_def, auto) |
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apply (rule theI2, fast) |
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apply (fast intro: inv_unique, fast) |
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done |
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lemma (in monoid) Units_l_inv_ex: |
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"x \<in> Units G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
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by (unfold Units_def) auto |
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lemma (in monoid) Units_r_inv_ex: |
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"x \<in> Units G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>" |
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by (unfold Units_def) auto |
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lemma (in monoid) Units_l_inv [simp]: |
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"x \<in> Units G ==> inv x \<otimes> x = \<one>" |
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apply (unfold Units_def m_inv_def, auto) |
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apply (rule theI2, fast) |
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apply (fast intro: inv_unique, fast) |
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done |
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lemma (in monoid) Units_r_inv [simp]: |
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"x \<in> Units G ==> x \<otimes> inv x = \<one>" |
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apply (unfold Units_def m_inv_def, auto) |
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apply (rule theI2, fast) |
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apply (fast intro: inv_unique, fast) |
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done |
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lemma (in monoid) Units_inv_Units [intro, simp]: |
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"x \<in> Units G ==> inv x \<in> Units G" |
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proof - |
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assume x: "x \<in> Units G" |
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show "inv x \<in> Units G" |
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by (auto simp add: Units_def |
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intro: Units_l_inv Units_r_inv x Units_closed [OF x]) |
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qed |
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lemma (in monoid) Units_l_cancel [simp]: |
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"[| x \<in> Units G; y \<in> carrier G; z \<in> carrier G |] ==> |
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(x \<otimes> y = x \<otimes> z) = (y = z)" |
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proof |
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assume eq: "x \<otimes> y = x \<otimes> z" |
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and G: "x \<in> Units G" "y \<in> carrier G" "z \<in> carrier G" |
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then have "(inv x \<otimes> x) \<otimes> y = (inv x \<otimes> x) \<otimes> z" |
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by (simp add: m_assoc Units_closed del: Units_l_inv) |
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with G show "y = z" by (simp add: Units_l_inv) |
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next |
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assume eq: "y = z" |
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and G: "x \<in> Units G" "y \<in> carrier G" "z \<in> carrier G" |
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then show "x \<otimes> y = x \<otimes> z" by simp |
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qed |
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lemma (in monoid) Units_inv_inv [simp]: |
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"x \<in> Units G ==> inv (inv x) = x" |
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proof - |
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assume x: "x \<in> Units G" |
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then have "inv x \<otimes> inv (inv x) = inv x \<otimes> x" by simp |
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with x show ?thesis by (simp add: Units_closed del: Units_l_inv Units_r_inv) |
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qed |
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lemma (in monoid) inv_inj_on_Units: |
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"inj_on (m_inv G) (Units G)" |
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proof (rule inj_onI) |
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fix x y |
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assume G: "x \<in> Units G" "y \<in> Units G" and eq: "inv x = inv y" |
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then have "inv (inv x) = inv (inv y)" by simp |
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with G show "x = y" by simp |
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qed |
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lemma (in monoid) Units_inv_comm: |
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assumes inv: "x \<otimes> y = \<one>" |
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and G: "x \<in> Units G" "y \<in> Units G" |
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shows "y \<otimes> x = \<one>" |
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proof - |
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from G have "x \<otimes> y \<otimes> x = x \<otimes> \<one>" by (auto simp add: inv Units_closed) |
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with G show ?thesis by (simp del: r_one add: m_assoc Units_closed) |
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qed |
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text {* Power *} |
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lemma (in monoid) nat_pow_closed [intro, simp]: |
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"x \<in> carrier G ==> x (^) (n::nat) \<in> carrier G" |
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by (induct n) (simp_all add: nat_pow_def) |
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lemma (in monoid) nat_pow_0 [simp]: |
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"x (^) (0::nat) = \<one>" |
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by (simp add: nat_pow_def) |
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lemma (in monoid) nat_pow_Suc [simp]: |
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"x (^) (Suc n) = x (^) n \<otimes> x" |
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by (simp add: nat_pow_def) |
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lemma (in monoid) nat_pow_one [simp]: |
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"\<one> (^) (n::nat) = \<one>" |
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by (induct n) simp_all |
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lemma (in monoid) nat_pow_mult: |
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"x \<in> carrier G ==> x (^) (n::nat) \<otimes> x (^) m = x (^) (n + m)" |
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by (induct m) (simp_all add: m_assoc [THEN sym]) |
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lemma (in monoid) nat_pow_pow: |
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"x \<in> carrier G ==> (x (^) n) (^) m = x (^) (n * m::nat)" |
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by (induct m) (simp, simp add: nat_pow_mult add_commute) |
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(* Jacobson defines submonoid here. *) |
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(* Jacobson defines the order of a monoid here. *) |
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subsection {* Groups *} |
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text {* |
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A group is a monoid all of whose elements are invertible. |
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*} |
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locale group = monoid + |
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assumes Units: "carrier G <= Units G" |
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lemma (in group) is_group: "group G" by (rule group_axioms) |
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theorem groupI: |
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fixes G (structure) |
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assumes m_closed [simp]: |
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"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G" |
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and one_closed [simp]: "\<one> \<in> carrier G" |
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and m_assoc: |
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"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
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(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
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and l_one [simp]: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" |
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and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
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shows "group G" |
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proof - |
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have l_cancel [simp]: |
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"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
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(x \<otimes> y = x \<otimes> z) = (y = z)" |
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proof |
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fix x y z |
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assume eq: "x \<otimes> y = x \<otimes> z" |
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and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
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with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G" |
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and l_inv: "x_inv \<otimes> x = \<one>" by fast |
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from G eq xG have "(x_inv \<otimes> x) \<otimes> y = (x_inv \<otimes> x) \<otimes> z" |
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by (simp add: m_assoc) |
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with G show "y = z" by (simp add: l_inv) |
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next |
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fix x y z |
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assume eq: "y = z" |
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and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
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then show "x \<otimes> y = x \<otimes> z" by simp |
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qed |
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have r_one: |
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"!!x. x \<in> carrier G ==> x \<otimes> \<one> = x" |
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proof - |
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fix x |
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assume x: "x \<in> carrier G" |
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with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G" |
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and l_inv: "x_inv \<otimes> x = \<one>" by fast |
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from x xG have "x_inv \<otimes> (x \<otimes> \<one>) = x_inv \<otimes> x" |
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by (simp add: m_assoc [symmetric] l_inv) |
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with x xG show "x \<otimes> \<one> = x" by simp |
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qed |
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have inv_ex: |
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"!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>" |
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proof - |
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fix x |
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assume x: "x \<in> carrier G" |
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with l_inv_ex obtain y where y: "y \<in> carrier G" |
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and l_inv: "y \<otimes> x = \<one>" by fast |
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from x y have "y \<otimes> (x \<otimes> y) = y \<otimes> \<one>" |
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13936 | 276 |
by (simp add: m_assoc [symmetric] l_inv r_one) |
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with x y have r_inv: "x \<otimes> y = \<one>" |
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by simp |
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from x y show "\<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>" |
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by (fast intro: l_inv r_inv) |
281 |
qed |
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then have carrier_subset_Units: "carrier G <= Units G" |
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by (unfold Units_def) fast |
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show ?thesis proof qed (auto simp: r_one m_assoc carrier_subset_Units) |
13936 | 285 |
qed |
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27698 | 287 |
lemma (in monoid) group_l_invI: |
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assumes l_inv_ex: |
14963 | 289 |
"!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
13936 | 290 |
shows "group G" |
291 |
by (rule groupI) (auto intro: m_assoc l_inv_ex) |
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292 |
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293 |
lemma (in group) Units_eq [simp]: |
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294 |
"Units G = carrier G" |
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proof |
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show "Units G <= carrier G" by fast |
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297 |
next |
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298 |
show "carrier G <= Units G" by (rule Units) |
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299 |
qed |
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300 |
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301 |
lemma (in group) inv_closed [intro, simp]: |
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302 |
"x \<in> carrier G ==> inv x \<in> carrier G" |
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using Units_inv_closed by simp |
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304 |
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19981 | 305 |
lemma (in group) l_inv_ex [simp]: |
306 |
"x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
|
307 |
using Units_l_inv_ex by simp |
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308 |
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309 |
lemma (in group) r_inv_ex [simp]: |
|
310 |
"x \<in> carrier G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>" |
|
311 |
using Units_r_inv_ex by simp |
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312 |
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14963 | 313 |
lemma (in group) l_inv [simp]: |
13936 | 314 |
"x \<in> carrier G ==> inv x \<otimes> x = \<one>" |
315 |
using Units_l_inv by simp |
|
13813 | 316 |
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0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
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317 |
|
13813 | 318 |
subsection {* Cancellation Laws and Basic Properties *} |
319 |
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320 |
lemma (in group) l_cancel [simp]: |
|
321 |
"[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
322 |
(x \<otimes> y = x \<otimes> z) = (y = z)" |
|
13936 | 323 |
using Units_l_inv by simp |
13940 | 324 |
|
14963 | 325 |
lemma (in group) r_inv [simp]: |
13813 | 326 |
"x \<in> carrier G ==> x \<otimes> inv x = \<one>" |
327 |
proof - |
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328 |
assume x: "x \<in> carrier G" |
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329 |
then have "inv x \<otimes> (x \<otimes> inv x) = inv x \<otimes> \<one>" |
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330 |
by (simp add: m_assoc [symmetric] l_inv) |
|
331 |
with x show ?thesis by (simp del: r_one) |
|
332 |
qed |
|
333 |
||
334 |
lemma (in group) r_cancel [simp]: |
|
335 |
"[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
336 |
(y \<otimes> x = z \<otimes> x) = (y = z)" |
|
337 |
proof |
|
338 |
assume eq: "y \<otimes> x = z \<otimes> x" |
|
14693 | 339 |
and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
13813 | 340 |
then have "y \<otimes> (x \<otimes> inv x) = z \<otimes> (x \<otimes> inv x)" |
27698 | 341 |
by (simp add: m_assoc [symmetric] del: r_inv Units_r_inv) |
14963 | 342 |
with G show "y = z" by simp |
13813 | 343 |
next |
344 |
assume eq: "y = z" |
|
14693 | 345 |
and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
13813 | 346 |
then show "y \<otimes> x = z \<otimes> x" by simp |
347 |
qed |
|
348 |
||
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First distributed version of Group and Ring theory.
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349 |
lemma (in group) inv_one [simp]: |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
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350 |
"inv \<one> = \<one>" |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
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|
351 |
proof - |
27698 | 352 |
have "inv \<one> = \<one> \<otimes> (inv \<one>)" by (simp del: r_inv Units_r_inv) |
14963 | 353 |
moreover have "... = \<one>" by simp |
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
354 |
finally show ?thesis . |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
355 |
qed |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
356 |
|
13813 | 357 |
lemma (in group) inv_inv [simp]: |
358 |
"x \<in> carrier G ==> inv (inv x) = x" |
|
13936 | 359 |
using Units_inv_inv by simp |
360 |
||
361 |
lemma (in group) inv_inj: |
|
362 |
"inj_on (m_inv G) (carrier G)" |
|
363 |
using inv_inj_on_Units by simp |
|
13813 | 364 |
|
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
365 |
lemma (in group) inv_mult_group: |
13813 | 366 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x" |
367 |
proof - |
|
14693 | 368 |
assume G: "x \<in> carrier G" "y \<in> carrier G" |
13813 | 369 |
then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)" |
14963 | 370 |
by (simp add: m_assoc l_inv) (simp add: m_assoc [symmetric]) |
27698 | 371 |
with G show ?thesis by (simp del: l_inv Units_l_inv) |
13813 | 372 |
qed |
373 |
||
13940 | 374 |
lemma (in group) inv_comm: |
375 |
"[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>" |
|
14693 | 376 |
by (rule Units_inv_comm) auto |
13940 | 377 |
|
13944 | 378 |
lemma (in group) inv_equality: |
13943 | 379 |
"[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y" |
380 |
apply (simp add: m_inv_def) |
|
381 |
apply (rule the_equality) |
|
14693 | 382 |
apply (simp add: inv_comm [of y x]) |
383 |
apply (rule r_cancel [THEN iffD1], auto) |
|
13943 | 384 |
done |
385 |
||
13936 | 386 |
text {* Power *} |
387 |
||
388 |
lemma (in group) int_pow_def2: |
|
389 |
"a (^) (z::int) = (if neg z then inv (a (^) (nat (-z))) else a (^) (nat z))" |
|
390 |
by (simp add: int_pow_def nat_pow_def Let_def) |
|
391 |
||
392 |
lemma (in group) int_pow_0 [simp]: |
|
393 |
"x (^) (0::int) = \<one>" |
|
394 |
by (simp add: int_pow_def2) |
|
395 |
||
396 |
lemma (in group) int_pow_one [simp]: |
|
397 |
"\<one> (^) (z::int) = \<one>" |
|
398 |
by (simp add: int_pow_def2) |
|
399 |
||
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
400 |
|
14963 | 401 |
subsection {* Subgroups *} |
13813 | 402 |
|
19783 | 403 |
locale subgroup = |
404 |
fixes H and G (structure) |
|
14963 | 405 |
assumes subset: "H \<subseteq> carrier G" |
406 |
and m_closed [intro, simp]: "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> H" |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
407 |
and one_closed [simp]: "\<one> \<in> H" |
14963 | 408 |
and m_inv_closed [intro,simp]: "x \<in> H \<Longrightarrow> inv x \<in> H" |
13813 | 409 |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
410 |
lemma (in subgroup) is_subgroup: |
26199 | 411 |
"subgroup H G" by (rule subgroup_axioms) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
412 |
|
13813 | 413 |
declare (in subgroup) group.intro [intro] |
13949
0ce528cd6f19
HOL-Algebra complete for release Isabelle2003 (modulo section headers).
ballarin
parents:
13944
diff
changeset
|
414 |
|
14963 | 415 |
lemma (in subgroup) mem_carrier [simp]: |
416 |
"x \<in> H \<Longrightarrow> x \<in> carrier G" |
|
417 |
using subset by blast |
|
13813 | 418 |
|
14963 | 419 |
lemma subgroup_imp_subset: |
420 |
"subgroup H G \<Longrightarrow> H \<subseteq> carrier G" |
|
421 |
by (rule subgroup.subset) |
|
422 |
||
423 |
lemma (in subgroup) subgroup_is_group [intro]: |
|
27611 | 424 |
assumes "group G" |
425 |
shows "group (G\<lparr>carrier := H\<rparr>)" |
|
426 |
proof - |
|
29237 | 427 |
interpret group G by fact |
27611 | 428 |
show ?thesis |
27698 | 429 |
apply (rule monoid.group_l_invI) |
430 |
apply (unfold_locales) [1] |
|
431 |
apply (auto intro: m_assoc l_inv mem_carrier) |
|
432 |
done |
|
27611 | 433 |
qed |
13813 | 434 |
|
435 |
text {* |
|
436 |
Since @{term H} is nonempty, it contains some element @{term x}. Since |
|
437 |
it is closed under inverse, it contains @{text "inv x"}. Since |
|
438 |
it is closed under product, it contains @{text "x \<otimes> inv x = \<one>"}. |
|
439 |
*} |
|
440 |
||
441 |
lemma (in group) one_in_subset: |
|
442 |
"[| H \<subseteq> carrier G; H \<noteq> {}; \<forall>a \<in> H. inv a \<in> H; \<forall>a\<in>H. \<forall>b\<in>H. a \<otimes> b \<in> H |] |
|
443 |
==> \<one> \<in> H" |
|
444 |
by (force simp add: l_inv) |
|
445 |
||
446 |
text {* A characterization of subgroups: closed, non-empty subset. *} |
|
447 |
||
448 |
lemma (in group) subgroupI: |
|
449 |
assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}" |
|
14963 | 450 |
and inv: "!!a. a \<in> H \<Longrightarrow> inv a \<in> H" |
451 |
and mult: "!!a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes> b \<in> H" |
|
13813 | 452 |
shows "subgroup H G" |
27714
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
453 |
proof (simp add: subgroup_def assms) |
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
454 |
show "\<one> \<in> H" by (rule one_in_subset) (auto simp only: assms) |
13813 | 455 |
qed |
456 |
||
13936 | 457 |
declare monoid.one_closed [iff] group.inv_closed [simp] |
458 |
monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp] |
|
13813 | 459 |
|
460 |
lemma subgroup_nonempty: |
|
461 |
"~ subgroup {} G" |
|
462 |
by (blast dest: subgroup.one_closed) |
|
463 |
||
464 |
lemma (in subgroup) finite_imp_card_positive: |
|
465 |
"finite (carrier G) ==> 0 < card H" |
|
466 |
proof (rule classical) |
|
14963 | 467 |
assume "finite (carrier G)" "~ 0 < card H" |
468 |
then have "finite H" by (blast intro: finite_subset [OF subset]) |
|
469 |
with prems have "subgroup {} G" by simp |
|
13813 | 470 |
with subgroup_nonempty show ?thesis by contradiction |
471 |
qed |
|
472 |
||
13936 | 473 |
(* |
474 |
lemma (in monoid) Units_subgroup: |
|
475 |
"subgroup (Units G) G" |
|
476 |
*) |
|
477 |
||
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
478 |
|
13813 | 479 |
subsection {* Direct Products *} |
480 |
||
14963 | 481 |
constdefs |
482 |
DirProd :: "_ \<Rightarrow> _ \<Rightarrow> ('a \<times> 'b) monoid" (infixr "\<times>\<times>" 80) |
|
483 |
"G \<times>\<times> H \<equiv> \<lparr>carrier = carrier G \<times> carrier H, |
|
484 |
mult = (\<lambda>(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')), |
|
485 |
one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>" |
|
13813 | 486 |
|
14963 | 487 |
lemma DirProd_monoid: |
27611 | 488 |
assumes "monoid G" and "monoid H" |
14963 | 489 |
shows "monoid (G \<times>\<times> H)" |
490 |
proof - |
|
29237 | 491 |
interpret G!: monoid G by fact |
492 |
interpret H!: monoid H by fact |
|
27714
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
493 |
from assms |
14963 | 494 |
show ?thesis by (unfold monoid_def DirProd_def, auto) |
495 |
qed |
|
13813 | 496 |
|
497 |
||
14963 | 498 |
text{*Does not use the previous result because it's easier just to use auto.*} |
499 |
lemma DirProd_group: |
|
27611 | 500 |
assumes "group G" and "group H" |
14963 | 501 |
shows "group (G \<times>\<times> H)" |
27611 | 502 |
proof - |
29237 | 503 |
interpret G!: group G by fact |
504 |
interpret H!: group H by fact |
|
27611 | 505 |
show ?thesis by (rule groupI) |
14963 | 506 |
(auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv |
507 |
simp add: DirProd_def) |
|
27611 | 508 |
qed |
13813 | 509 |
|
14963 | 510 |
lemma carrier_DirProd [simp]: |
511 |
"carrier (G \<times>\<times> H) = carrier G \<times> carrier H" |
|
512 |
by (simp add: DirProd_def) |
|
13944 | 513 |
|
14963 | 514 |
lemma one_DirProd [simp]: |
515 |
"\<one>\<^bsub>G \<times>\<times> H\<^esub> = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)" |
|
516 |
by (simp add: DirProd_def) |
|
13944 | 517 |
|
14963 | 518 |
lemma mult_DirProd [simp]: |
519 |
"(g, h) \<otimes>\<^bsub>(G \<times>\<times> H)\<^esub> (g', h') = (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')" |
|
520 |
by (simp add: DirProd_def) |
|
13944 | 521 |
|
14963 | 522 |
lemma inv_DirProd [simp]: |
27611 | 523 |
assumes "group G" and "group H" |
13944 | 524 |
assumes g: "g \<in> carrier G" |
525 |
and h: "h \<in> carrier H" |
|
14963 | 526 |
shows "m_inv (G \<times>\<times> H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)" |
27611 | 527 |
proof - |
29237 | 528 |
interpret G!: group G by fact |
529 |
interpret H!: group H by fact |
|
530 |
interpret Prod!: group "G \<times>\<times> H" |
|
27714
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
531 |
by (auto intro: DirProd_group group.intro group.axioms assms) |
14963 | 532 |
show ?thesis by (simp add: Prod.inv_equality g h) |
533 |
qed |
|
27698 | 534 |
|
14963 | 535 |
|
536 |
subsection {* Homomorphisms and Isomorphisms *} |
|
13813 | 537 |
|
14651 | 538 |
constdefs (structure G and H) |
539 |
hom :: "_ => _ => ('a => 'b) set" |
|
13813 | 540 |
"hom G H == |
541 |
{h. h \<in> carrier G -> carrier H & |
|
14693 | 542 |
(\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y)}" |
13813 | 543 |
|
14761 | 544 |
lemma (in group) hom_compose: |
545 |
"[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I" |
|
546 |
apply (auto simp add: hom_def funcset_compose) |
|
547 |
apply (simp add: compose_def funcset_mem) |
|
13943 | 548 |
done |
549 |
||
14803 | 550 |
constdefs |
551 |
iso :: "_ => _ => ('a => 'b) set" (infixr "\<cong>" 60) |
|
552 |
"G \<cong> H == {h. h \<in> hom G H & bij_betw h (carrier G) (carrier H)}" |
|
14761 | 553 |
|
14803 | 554 |
lemma iso_refl: "(%x. x) \<in> G \<cong> G" |
14761 | 555 |
by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) |
556 |
||
557 |
lemma (in group) iso_sym: |
|
14803 | 558 |
"h \<in> G \<cong> H \<Longrightarrow> Inv (carrier G) h \<in> H \<cong> G" |
14761 | 559 |
apply (simp add: iso_def bij_betw_Inv) |
560 |
apply (subgoal_tac "Inv (carrier G) h \<in> carrier H \<rightarrow> carrier G") |
|
561 |
prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_Inv]) |
|
562 |
apply (simp add: hom_def bij_betw_def Inv_f_eq funcset_mem f_Inv_f) |
|
563 |
done |
|
564 |
||
565 |
lemma (in group) iso_trans: |
|
14803 | 566 |
"[|h \<in> G \<cong> H; i \<in> H \<cong> I|] ==> (compose (carrier G) i h) \<in> G \<cong> I" |
14761 | 567 |
by (auto simp add: iso_def hom_compose bij_betw_compose) |
568 |
||
14963 | 569 |
lemma DirProd_commute_iso: |
570 |
shows "(\<lambda>(x,y). (y,x)) \<in> (G \<times>\<times> H) \<cong> (H \<times>\<times> G)" |
|
14761 | 571 |
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) |
572 |
||
14963 | 573 |
lemma DirProd_assoc_iso: |
574 |
shows "(\<lambda>(x,y,z). (x,(y,z))) \<in> (G \<times>\<times> H \<times>\<times> I) \<cong> (G \<times>\<times> (H \<times>\<times> I))" |
|
14761 | 575 |
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) |
576 |
||
577 |
||
14963 | 578 |
text{*Basis for homomorphism proofs: we assume two groups @{term G} and |
15076
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14963
diff
changeset
|
579 |
@{term H}, with a homomorphism @{term h} between them*} |
29237 | 580 |
locale group_hom = G: group G + H: group H for G (structure) and H (structure) + |
581 |
fixes h |
|
13813 | 582 |
assumes homh: "h \<in> hom G H" |
29240 | 583 |
|
584 |
lemma (in group_hom) hom_mult [simp]: |
|
585 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y" |
|
586 |
proof - |
|
587 |
assume "x \<in> carrier G" "y \<in> carrier G" |
|
588 |
with homh [unfolded hom_def] show ?thesis by simp |
|
589 |
qed |
|
590 |
||
591 |
lemma (in group_hom) hom_closed [simp]: |
|
592 |
"x \<in> carrier G ==> h x \<in> carrier H" |
|
593 |
proof - |
|
594 |
assume "x \<in> carrier G" |
|
595 |
with homh [unfolded hom_def] show ?thesis by (auto simp add: funcset_mem) |
|
596 |
qed |
|
13813 | 597 |
|
598 |
lemma (in group_hom) one_closed [simp]: |
|
599 |
"h \<one> \<in> carrier H" |
|
600 |
by simp |
|
601 |
||
602 |
lemma (in group_hom) hom_one [simp]: |
|
14693 | 603 |
"h \<one> = \<one>\<^bsub>H\<^esub>" |
13813 | 604 |
proof - |
15076
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14963
diff
changeset
|
605 |
have "h \<one> \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = h \<one> \<otimes>\<^bsub>H\<^esub> h \<one>" |
13813 | 606 |
by (simp add: hom_mult [symmetric] del: hom_mult) |
607 |
then show ?thesis by (simp del: r_one) |
|
608 |
qed |
|
609 |
||
610 |
lemma (in group_hom) inv_closed [simp]: |
|
611 |
"x \<in> carrier G ==> h (inv x) \<in> carrier H" |
|
612 |
by simp |
|
613 |
||
614 |
lemma (in group_hom) hom_inv [simp]: |
|
14693 | 615 |
"x \<in> carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)" |
13813 | 616 |
proof - |
617 |
assume x: "x \<in> carrier G" |
|
14693 | 618 |
then have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = \<one>\<^bsub>H\<^esub>" |
14963 | 619 |
by (simp add: hom_mult [symmetric] del: hom_mult) |
14693 | 620 |
also from x have "... = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" |
14963 | 621 |
by (simp add: hom_mult [symmetric] del: hom_mult) |
14693 | 622 |
finally have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" . |
27698 | 623 |
with x show ?thesis by (simp del: H.r_inv H.Units_r_inv) |
13813 | 624 |
qed |
625 |
||
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
626 |
|
13949
0ce528cd6f19
HOL-Algebra complete for release Isabelle2003 (modulo section headers).
ballarin
parents:
13944
diff
changeset
|
627 |
subsection {* Commutative Structures *} |
13936 | 628 |
|
629 |
text {* |
|
630 |
Naming convention: multiplicative structures that are commutative |
|
631 |
are called \emph{commutative}, additive structures are called |
|
632 |
\emph{Abelian}. |
|
633 |
*} |
|
13813 | 634 |
|
14963 | 635 |
locale comm_monoid = monoid + |
636 |
assumes m_comm: "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x" |
|
13813 | 637 |
|
14963 | 638 |
lemma (in comm_monoid) m_lcomm: |
639 |
"\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> |
|
13813 | 640 |
x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)" |
641 |
proof - |
|
14693 | 642 |
assume xyz: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
13813 | 643 |
from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc) |
644 |
also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm) |
|
645 |
also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc) |
|
646 |
finally show ?thesis . |
|
647 |
qed |
|
648 |
||
14963 | 649 |
lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm |
13813 | 650 |
|
13936 | 651 |
lemma comm_monoidI: |
19783 | 652 |
fixes G (structure) |
13936 | 653 |
assumes m_closed: |
14693 | 654 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G" |
655 |
and one_closed: "\<one> \<in> carrier G" |
|
13936 | 656 |
and m_assoc: |
657 |
"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
14693 | 658 |
(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
659 |
and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" |
|
13936 | 660 |
and m_comm: |
14693 | 661 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x" |
13936 | 662 |
shows "comm_monoid G" |
663 |
using l_one |
|
14963 | 664 |
by (auto intro!: comm_monoid.intro comm_monoid_axioms.intro monoid.intro |
27714
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
665 |
intro: assms simp: m_closed one_closed m_comm) |
13817 | 666 |
|
13936 | 667 |
lemma (in monoid) monoid_comm_monoidI: |
668 |
assumes m_comm: |
|
14693 | 669 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x" |
13936 | 670 |
shows "comm_monoid G" |
671 |
by (rule comm_monoidI) (auto intro: m_assoc m_comm) |
|
14963 | 672 |
|
14693 | 673 |
(*lemma (in comm_monoid) r_one [simp]: |
13817 | 674 |
"x \<in> carrier G ==> x \<otimes> \<one> = x" |
675 |
proof - |
|
676 |
assume G: "x \<in> carrier G" |
|
677 |
then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm) |
|
678 |
also from G have "... = x" by simp |
|
679 |
finally show ?thesis . |
|
14693 | 680 |
qed*) |
14963 | 681 |
|
13936 | 682 |
lemma (in comm_monoid) nat_pow_distr: |
683 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> |
|
684 |
(x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n" |
|
685 |
by (induct n) (simp, simp add: m_ac) |
|
686 |
||
687 |
locale comm_group = comm_monoid + group |
|
688 |
||
689 |
lemma (in group) group_comm_groupI: |
|
690 |
assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> |
|
14693 | 691 |
x \<otimes> y = y \<otimes> x" |
13936 | 692 |
shows "comm_group G" |
28823 | 693 |
proof qed (simp_all add: m_comm) |
13817 | 694 |
|
13936 | 695 |
lemma comm_groupI: |
19783 | 696 |
fixes G (structure) |
13936 | 697 |
assumes m_closed: |
14693 | 698 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G" |
699 |
and one_closed: "\<one> \<in> carrier G" |
|
13936 | 700 |
and m_assoc: |
701 |
"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
14693 | 702 |
(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
13936 | 703 |
and m_comm: |
14693 | 704 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x" |
705 |
and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" |
|
14963 | 706 |
and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
13936 | 707 |
shows "comm_group G" |
27714
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
708 |
by (fast intro: group.group_comm_groupI groupI assms) |
13936 | 709 |
|
710 |
lemma (in comm_group) inv_mult: |
|
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
711 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y" |
13936 | 712 |
by (simp add: m_ac inv_mult_group) |
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
713 |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
714 |
|
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
715 |
subsection {* The Lattice of Subgroups of a Group *} |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
716 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
717 |
text_raw {* \label{sec:subgroup-lattice} *} |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
718 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
719 |
theorem (in group) subgroups_partial_order: |
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27698
diff
changeset
|
720 |
"partial_order (| carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> |)" |
28823 | 721 |
proof qed simp_all |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
722 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
723 |
lemma (in group) subgroup_self: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
724 |
"subgroup (carrier G) G" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
725 |
by (rule subgroupI) auto |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
726 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
727 |
lemma (in group) subgroup_imp_group: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
728 |
"subgroup H G ==> group (G(| carrier := H |))" |
26199 | 729 |
by (erule subgroup.subgroup_is_group) (rule group_axioms) |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
730 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
731 |
lemma (in group) is_monoid [intro, simp]: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
732 |
"monoid G" |
14963 | 733 |
by (auto intro: monoid.intro m_assoc) |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
734 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
735 |
lemma (in group) subgroup_inv_equality: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
736 |
"[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
737 |
apply (rule_tac inv_equality [THEN sym]) |
14761 | 738 |
apply (rule group.l_inv [OF subgroup_imp_group, simplified], assumption+) |
739 |
apply (rule subsetD [OF subgroup.subset], assumption+) |
|
740 |
apply (rule subsetD [OF subgroup.subset], assumption) |
|
741 |
apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified], assumption+) |
|
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
742 |
done |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
743 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
744 |
theorem (in group) subgroups_Inter: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
745 |
assumes subgr: "(!!H. H \<in> A ==> subgroup H G)" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
746 |
and not_empty: "A ~= {}" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
747 |
shows "subgroup (\<Inter>A) G" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
748 |
proof (rule subgroupI) |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
749 |
from subgr [THEN subgroup.subset] and not_empty |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
750 |
show "\<Inter>A \<subseteq> carrier G" by blast |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
751 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
752 |
from subgr [THEN subgroup.one_closed] |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
753 |
show "\<Inter>A ~= {}" by blast |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
754 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
755 |
fix x assume "x \<in> \<Inter>A" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
756 |
with subgr [THEN subgroup.m_inv_closed] |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
757 |
show "inv x \<in> \<Inter>A" by blast |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
758 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
759 |
fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
760 |
with subgr [THEN subgroup.m_closed] |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
761 |
show "x \<otimes> y \<in> \<Inter>A" by blast |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
762 |
qed |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
763 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
764 |
theorem (in group) subgroups_complete_lattice: |
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27698
diff
changeset
|
765 |
"complete_lattice (| carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> |)" |
22063
717425609192
Reverted to structure representation with records.
ballarin
parents:
21041
diff
changeset
|
766 |
(is "complete_lattice ?L") |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
767 |
proof (rule partial_order.complete_lattice_criterion1) |
22063
717425609192
Reverted to structure representation with records.
ballarin
parents:
21041
diff
changeset
|
768 |
show "partial_order ?L" by (rule subgroups_partial_order) |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
769 |
next |
26805
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
770 |
show "\<exists>G. greatest ?L G (carrier ?L)" |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
771 |
proof |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
772 |
show "greatest ?L (carrier G) (carrier ?L)" |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
773 |
by (unfold greatest_def) |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
774 |
(simp add: subgroup.subset subgroup_self) |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
775 |
qed |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
776 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
777 |
fix A |
22063
717425609192
Reverted to structure representation with records.
ballarin
parents:
21041
diff
changeset
|
778 |
assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}" |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
779 |
then have Int_subgroup: "subgroup (\<Inter>A) G" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
780 |
by (fastsimp intro: subgroups_Inter) |
26805
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
781 |
show "\<exists>I. greatest ?L I (Lower ?L A)" |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
782 |
proof |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
783 |
show "greatest ?L (\<Inter>A) (Lower ?L A)" |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
784 |
(is "greatest _ ?Int _") |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
785 |
proof (rule greatest_LowerI) |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
786 |
fix H |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
787 |
assume H: "H \<in> A" |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
788 |
with L have subgroupH: "subgroup H G" by auto |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
789 |
from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H") |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
790 |
by (rule subgroup_imp_group) |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
791 |
from groupH have monoidH: "monoid ?H" |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
792 |
by (rule group.is_monoid) |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
793 |
from H have Int_subset: "?Int \<subseteq> H" by fastsimp |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
794 |
then show "le ?L ?Int H" by simp |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
795 |
next |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
796 |
fix H |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
797 |
assume H: "H \<in> Lower ?L A" |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
798 |
with L Int_subgroup show "le ?L H ?Int" |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
799 |
by (fastsimp simp: Lower_def intro: Inter_greatest) |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
800 |
next |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
801 |
show "A \<subseteq> carrier ?L" by (rule L) |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
802 |
next |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
803 |
show "?Int \<in> carrier ?L" by simp (rule Int_subgroup) |
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset
|
804 |
qed |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
805 |
qed |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
806 |
qed |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
807 |
|
13813 | 808 |
end |