author | paulson <lp15@cam.ac.uk> |
Sun, 08 Jul 2018 23:35:33 +0100 | |
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parent 68555 | 22d51874f37d |
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permissions | -rw-r--r-- |
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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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With additional contributions from Martin Baillon and Paulo EmÃlio de Vilhena. |
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*) |
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theory Group |
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imports Complete_Lattice "HOL-Library.FuncSet" |
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begin |
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section \<open>Monoids and Groups\<close> |
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subsection \<open>Definitions\<close> |
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text \<open> |
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Definitions follow @{cite "Jacobson:1985"}. |
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\<close> |
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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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definition |
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m_inv :: "('a, 'b) monoid_scheme => 'a => 'a" ("inv\<index> _" [81] 80) |
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where "inv\<^bsub>G\<^esub> x = (THE y. y \<in> carrier G \<and> x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> \<and> y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>)" |
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definition |
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Units :: "_ => 'a set" |
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\<comment> \<open>The set of invertible elements\<close> |
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where "Units G = {y. y \<in> carrier G \<and> (\<exists>x \<in> carrier G. x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> \<and> y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>)}" |
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consts |
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pow :: "[('a, 'm) monoid_scheme, 'a, 'b::semiring_1] => 'a" (infixr "[^]\<index>" 75) |
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overloading nat_pow == "pow :: [_, 'a, nat] => 'a" |
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begin |
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definition "nat_pow G a n = rec_nat \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) n" |
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end |
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overloading int_pow == "pow :: [_, 'a, int] => 'a" |
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begin |
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definition "int_pow G a z = |
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(let p = rec_nat \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) |
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in if z < 0 then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z))" |
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end |
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lemma int_pow_int: "x [^]\<^bsub>G\<^esub> (int n) = x [^]\<^bsub>G\<^esub> n" |
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by(simp add: int_pow_def nat_pow_def) |
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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) one_unique: |
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assumes "u \<in> carrier G" |
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and "\<And>x. x \<in> carrier G \<Longrightarrow> u \<otimes> x = x" |
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shows "u = \<one>" |
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using assms(2)[OF one_closed] r_one[OF assms(1)] by simp |
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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 [simp, intro]: |
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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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by simp (metis m_assoc m_closed) |
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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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by (metis (full_types) Units_closed Units_inv_closed Units_l_inv Units_r_inv_ex inv_unique) |
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lemma (in monoid) inv_one [simp]: |
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"inv \<one> = \<one>" |
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by (metis Units_one_closed Units_r_inv l_one monoid.Units_inv_closed monoid_axioms) |
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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 |
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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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lemma (in monoid) carrier_not_empty: "carrier G \<noteq> {}" |
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by auto |
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text \<open>Power\<close> |
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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_comm: |
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"x \<in> carrier G \<Longrightarrow> (x [^] (n::nat)) \<otimes> (x [^] (m :: nat)) = (x [^] m) \<otimes> (x [^] n)" |
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using nat_pow_mult[of x n m] nat_pow_mult[of x m n] by (simp add: add.commute) |
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lemma (in monoid) nat_pow_Suc2: |
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"x \<in> carrier G \<Longrightarrow> x [^] (Suc n) = x \<otimes> (x [^] n)" |
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using nat_pow_mult[of x 1 n] Suc_eq_plus1[of n] |
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by (metis One_nat_def Suc_eq_plus1_left l_one nat.rec(1) nat_pow_Suc nat_pow_def) |
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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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lemma (in monoid) nat_pow_consistent: |
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"x [^] (n :: nat) = x [^]\<^bsub>(G \<lparr> carrier := H \<rparr>)\<^esub> n" |
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unfolding nat_pow_def by simp |
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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 \<open>Groups\<close> |
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text \<open> |
13936 | 245 |
A group is a monoid all of whose elements are invertible. |
61382 | 246 |
\<close> |
13936 | 247 |
|
248 |
locale group = monoid + |
|
249 |
assumes Units: "carrier G <= Units G" |
|
250 |
||
26199 | 251 |
lemma (in group) is_group: "group G" by (rule group_axioms) |
14761 | 252 |
|
13936 | 253 |
theorem groupI: |
19783 | 254 |
fixes G (structure) |
13936 | 255 |
assumes m_closed [simp]: |
14693 | 256 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G" |
257 |
and one_closed [simp]: "\<one> \<in> carrier G" |
|
13936 | 258 |
and m_assoc: |
259 |
"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
14693 | 260 |
(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
261 |
and l_one [simp]: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" |
|
14963 | 262 |
and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
13936 | 263 |
shows "group G" |
264 |
proof - |
|
265 |
have l_cancel [simp]: |
|
266 |
"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
14693 | 267 |
(x \<otimes> y = x \<otimes> z) = (y = z)" |
13936 | 268 |
proof |
269 |
fix x y z |
|
14693 | 270 |
assume eq: "x \<otimes> y = x \<otimes> z" |
271 |
and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
|
13936 | 272 |
with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G" |
14693 | 273 |
and l_inv: "x_inv \<otimes> x = \<one>" by fast |
274 |
from G eq xG have "(x_inv \<otimes> x) \<otimes> y = (x_inv \<otimes> x) \<otimes> z" |
|
13936 | 275 |
by (simp add: m_assoc) |
276 |
with G show "y = z" by (simp add: l_inv) |
|
277 |
next |
|
278 |
fix x y z |
|
279 |
assume eq: "y = z" |
|
14693 | 280 |
and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
281 |
then show "x \<otimes> y = x \<otimes> z" by simp |
|
13936 | 282 |
qed |
283 |
have r_one: |
|
14693 | 284 |
"!!x. x \<in> carrier G ==> x \<otimes> \<one> = x" |
13936 | 285 |
proof - |
286 |
fix x |
|
287 |
assume x: "x \<in> carrier G" |
|
288 |
with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G" |
|
14693 | 289 |
and l_inv: "x_inv \<otimes> x = \<one>" by fast |
290 |
from x xG have "x_inv \<otimes> (x \<otimes> \<one>) = x_inv \<otimes> x" |
|
13936 | 291 |
by (simp add: m_assoc [symmetric] l_inv) |
14693 | 292 |
with x xG show "x \<otimes> \<one> = x" by simp |
13936 | 293 |
qed |
294 |
have inv_ex: |
|
67091 | 295 |
"\<And>x. x \<in> carrier G \<Longrightarrow> \<exists>y \<in> carrier G. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" |
13936 | 296 |
proof - |
297 |
fix x |
|
298 |
assume x: "x \<in> carrier G" |
|
299 |
with l_inv_ex obtain y where y: "y \<in> carrier G" |
|
14693 | 300 |
and l_inv: "y \<otimes> x = \<one>" by fast |
301 |
from x y have "y \<otimes> (x \<otimes> y) = y \<otimes> \<one>" |
|
13936 | 302 |
by (simp add: m_assoc [symmetric] l_inv r_one) |
14693 | 303 |
with x y have r_inv: "x \<otimes> y = \<one>" |
13936 | 304 |
by simp |
67091 | 305 |
from x y show "\<exists>y \<in> carrier G. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" |
13936 | 306 |
by (fast intro: l_inv r_inv) |
307 |
qed |
|
67091 | 308 |
then have carrier_subset_Units: "carrier G \<subseteq> Units G" |
13936 | 309 |
by (unfold Units_def) fast |
61169 | 310 |
show ?thesis |
311 |
by standard (auto simp: r_one m_assoc carrier_subset_Units) |
|
13936 | 312 |
qed |
313 |
||
27698 | 314 |
lemma (in monoid) group_l_invI: |
13936 | 315 |
assumes l_inv_ex: |
14963 | 316 |
"!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
13936 | 317 |
shows "group G" |
318 |
by (rule groupI) (auto intro: m_assoc l_inv_ex) |
|
319 |
||
320 |
lemma (in group) Units_eq [simp]: |
|
321 |
"Units G = carrier G" |
|
322 |
proof |
|
67091 | 323 |
show "Units G \<subseteq> carrier G" by fast |
13936 | 324 |
next |
67091 | 325 |
show "carrier G \<subseteq> Units G" by (rule Units) |
13936 | 326 |
qed |
327 |
||
328 |
lemma (in group) inv_closed [intro, simp]: |
|
329 |
"x \<in> carrier G ==> inv x \<in> carrier G" |
|
330 |
using Units_inv_closed by simp |
|
331 |
||
19981 | 332 |
lemma (in group) l_inv_ex [simp]: |
333 |
"x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
|
334 |
using Units_l_inv_ex by simp |
|
335 |
||
336 |
lemma (in group) r_inv_ex [simp]: |
|
337 |
"x \<in> carrier G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>" |
|
338 |
using Units_r_inv_ex by simp |
|
339 |
||
14963 | 340 |
lemma (in group) l_inv [simp]: |
13936 | 341 |
"x \<in> carrier G ==> inv x \<otimes> x = \<one>" |
68399
0b71d08528f0
resolution of name clashes in Algebra
paulson <lp15@cam.ac.uk>
parents:
68188
diff
changeset
|
342 |
by simp |
13813 | 343 |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
344 |
|
61382 | 345 |
subsection \<open>Cancellation Laws and Basic Properties\<close> |
13813 | 346 |
|
14963 | 347 |
lemma (in group) r_inv [simp]: |
13813 | 348 |
"x \<in> carrier G ==> x \<otimes> inv x = \<one>" |
68399
0b71d08528f0
resolution of name clashes in Algebra
paulson <lp15@cam.ac.uk>
parents:
68188
diff
changeset
|
349 |
by simp |
13813 | 350 |
|
68399
0b71d08528f0
resolution of name clashes in Algebra
paulson <lp15@cam.ac.uk>
parents:
68188
diff
changeset
|
351 |
lemma (in group) right_cancel [simp]: |
13813 | 352 |
"[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
353 |
(y \<otimes> x = z \<otimes> x) = (y = z)" |
|
68399
0b71d08528f0
resolution of name clashes in Algebra
paulson <lp15@cam.ac.uk>
parents:
68188
diff
changeset
|
354 |
by (metis inv_closed m_assoc r_inv r_one) |
13813 | 355 |
|
356 |
lemma (in group) inv_inv [simp]: |
|
357 |
"x \<in> carrier G ==> inv (inv x) = x" |
|
13936 | 358 |
using Units_inv_inv by simp |
359 |
||
360 |
lemma (in group) inv_inj: |
|
361 |
"inj_on (m_inv G) (carrier G)" |
|
362 |
using inv_inj_on_Units by simp |
|
13813 | 363 |
|
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
364 |
lemma (in group) inv_mult_group: |
13813 | 365 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x" |
366 |
proof - |
|
14693 | 367 |
assume G: "x \<in> carrier G" "y \<in> carrier G" |
13813 | 368 |
then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)" |
44472 | 369 |
by (simp add: m_assoc) (simp add: m_assoc [symmetric]) |
27698 | 370 |
with G show ?thesis by (simp del: l_inv Units_l_inv) |
13813 | 371 |
qed |
372 |
||
13940 | 373 |
lemma (in group) inv_comm: |
374 |
"[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>" |
|
14693 | 375 |
by (rule Units_inv_comm) auto |
13940 | 376 |
|
13944 | 377 |
lemma (in group) inv_equality: |
13943 | 378 |
"[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y" |
68399
0b71d08528f0
resolution of name clashes in Algebra
paulson <lp15@cam.ac.uk>
parents:
68188
diff
changeset
|
379 |
using inv_unique r_inv by blast |
13943 | 380 |
|
57271 | 381 |
(* Contributed by Joachim Breitner *) |
382 |
lemma (in group) inv_solve_left: |
|
383 |
"\<lbrakk> a \<in> carrier G; b \<in> carrier G; c \<in> carrier G \<rbrakk> \<Longrightarrow> a = inv b \<otimes> c \<longleftrightarrow> c = b \<otimes> a" |
|
384 |
by (metis inv_equality l_inv_ex l_one m_assoc r_inv) |
|
385 |
lemma (in group) inv_solve_right: |
|
386 |
"\<lbrakk> a \<in> carrier G; b \<in> carrier G; c \<in> carrier G \<rbrakk> \<Longrightarrow> a = b \<otimes> inv c \<longleftrightarrow> b = a \<otimes> c" |
|
387 |
by (metis inv_equality l_inv_ex l_one m_assoc r_inv) |
|
388 |
||
61382 | 389 |
text \<open>Power\<close> |
13936 | 390 |
|
391 |
lemma (in group) int_pow_def2: |
|
67341
df79ef3b3a41
Renamed (^) to [^] in preparation of the move from "op X" to (X)
nipkow
parents:
67091
diff
changeset
|
392 |
"a [^] (z::int) = (if z < 0 then inv (a [^] (nat (-z))) else a [^] (nat z))" |
13936 | 393 |
by (simp add: int_pow_def nat_pow_def Let_def) |
394 |
||
395 |
lemma (in group) int_pow_0 [simp]: |
|
67341
df79ef3b3a41
Renamed (^) to [^] in preparation of the move from "op X" to (X)
nipkow
parents:
67091
diff
changeset
|
396 |
"x [^] (0::int) = \<one>" |
13936 | 397 |
by (simp add: int_pow_def2) |
398 |
||
399 |
lemma (in group) int_pow_one [simp]: |
|
67341
df79ef3b3a41
Renamed (^) to [^] in preparation of the move from "op X" to (X)
nipkow
parents:
67091
diff
changeset
|
400 |
"\<one> [^] (z::int) = \<one>" |
13936 | 401 |
by (simp add: int_pow_def2) |
402 |
||
57271 | 403 |
(* The following are contributed by Joachim Breitner *) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
404 |
|
57271 | 405 |
lemma (in group) int_pow_closed [intro, simp]: |
67341
df79ef3b3a41
Renamed (^) to [^] in preparation of the move from "op X" to (X)
nipkow
parents:
67091
diff
changeset
|
406 |
"x \<in> carrier G ==> x [^] (i::int) \<in> carrier G" |
57271 | 407 |
by (simp add: int_pow_def2) |
408 |
||
409 |
lemma (in group) int_pow_1 [simp]: |
|
67341
df79ef3b3a41
Renamed (^) to [^] in preparation of the move from "op X" to (X)
nipkow
parents:
67091
diff
changeset
|
410 |
"x \<in> carrier G \<Longrightarrow> x [^] (1::int) = x" |
57271 | 411 |
by (simp add: int_pow_def2) |
412 |
||
413 |
lemma (in group) int_pow_neg: |
|
67341
df79ef3b3a41
Renamed (^) to [^] in preparation of the move from "op X" to (X)
nipkow
parents:
67091
diff
changeset
|
414 |
"x \<in> carrier G \<Longrightarrow> x [^] (-i::int) = inv (x [^] i)" |
57271 | 415 |
by (simp add: int_pow_def2) |
416 |
||
417 |
lemma (in group) int_pow_mult: |
|
67341
df79ef3b3a41
Renamed (^) to [^] in preparation of the move from "op X" to (X)
nipkow
parents:
67091
diff
changeset
|
418 |
"x \<in> carrier G \<Longrightarrow> x [^] (i + j::int) = x [^] i \<otimes> x [^] j" |
57271 | 419 |
proof - |
420 |
have [simp]: "-i - j = -j - i" by simp |
|
67613 | 421 |
assume "x \<in> carrier G" then |
57271 | 422 |
show ?thesis |
423 |
by (auto simp add: int_pow_def2 inv_solve_left inv_solve_right nat_add_distrib [symmetric] nat_pow_mult ) |
|
424 |
qed |
|
425 |
||
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
426 |
lemma (in group) nat_pow_inv: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
427 |
"x \<in> carrier G \<Longrightarrow> (inv x) [^] (i :: nat) = inv (x [^] i)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
428 |
proof (induction i) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
429 |
case 0 thus ?case by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
430 |
next |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
431 |
case (Suc i) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
432 |
have "(inv x) [^] Suc i = ((inv x) [^] i) \<otimes> inv x" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
433 |
by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
434 |
also have " ... = (inv (x [^] i)) \<otimes> inv x" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
435 |
by (simp add: Suc.IH Suc.prems) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
436 |
also have " ... = inv (x \<otimes> (x [^] i))" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
437 |
using inv_mult_group[OF Suc.prems nat_pow_closed[OF Suc.prems, of i]] by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
438 |
also have " ... = inv (x [^] (Suc i))" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
439 |
using Suc.prems nat_pow_Suc2 by auto |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
440 |
finally show ?case . |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
441 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
442 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
443 |
lemma (in group) int_pow_inv: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
444 |
"x \<in> carrier G \<Longrightarrow> (inv x) [^] (i :: int) = inv (x [^] i)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
445 |
by (simp add: nat_pow_inv int_pow_def2) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
446 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
447 |
lemma (in group) int_pow_pow: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
448 |
assumes "x \<in> carrier G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
449 |
shows "(x [^] (n :: int)) [^] (m :: int) = x [^] (n * m :: int)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
450 |
proof (cases) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
451 |
assume n_ge: "n \<ge> 0" thus ?thesis |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
452 |
proof (cases) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
453 |
assume m_ge: "m \<ge> 0" thus ?thesis |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
454 |
using n_ge nat_pow_pow[OF assms, of "nat n" "nat m"] int_pow_def2 |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
455 |
by (simp add: mult_less_0_iff nat_mult_distrib) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
456 |
next |
68605 | 457 |
assume m_lt: "\<not> m \<ge> 0" |
458 |
with n_ge show ?thesis |
|
459 |
apply (simp add: int_pow_def2 mult_less_0_iff) |
|
460 |
by (metis assms mult_minus_right n_ge nat_mult_distrib nat_pow_pow) |
|
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
461 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
462 |
next |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
463 |
assume n_lt: "\<not> n \<ge> 0" thus ?thesis |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
464 |
proof (cases) |
68605 | 465 |
assume m_ge: "m \<ge> 0" |
466 |
have "inv x [^] (nat m * nat (- n)) = inv x [^] nat (- (m * n))" |
|
467 |
by (metis (full_types) m_ge mult_minus_right nat_mult_distrib) |
|
468 |
with m_ge n_lt show ?thesis |
|
469 |
by (simp add: int_pow_def2 mult_less_0_iff assms mult.commute nat_pow_inv nat_pow_pow) |
|
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
470 |
next |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
471 |
assume m_lt: "\<not> m \<ge> 0" thus ?thesis |
68605 | 472 |
using n_lt by (auto simp: int_pow_def2 mult_less_0_iff assms nat_mult_distrib_neg nat_pow_inv nat_pow_pow) |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
473 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
474 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
475 |
|
61628 | 476 |
lemma (in group) int_pow_diff: |
67341
df79ef3b3a41
Renamed (^) to [^] in preparation of the move from "op X" to (X)
nipkow
parents:
67091
diff
changeset
|
477 |
"x \<in> carrier G \<Longrightarrow> x [^] (n - m :: int) = x [^] n \<otimes> inv (x [^] m)" |
61628 | 478 |
by(simp only: diff_conv_add_uminus int_pow_mult int_pow_neg) |
479 |
||
480 |
lemma (in group) inj_on_multc: "c \<in> carrier G \<Longrightarrow> inj_on (\<lambda>x. x \<otimes> c) (carrier G)" |
|
481 |
by(simp add: inj_on_def) |
|
482 |
||
483 |
lemma (in group) inj_on_cmult: "c \<in> carrier G \<Longrightarrow> inj_on (\<lambda>x. c \<otimes> x) (carrier G)" |
|
484 |
by(simp add: inj_on_def) |
|
485 |
||
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
486 |
(*Following subsection contributed by Martin Baillon*) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
487 |
subsection \<open>Submonoids\<close> |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
488 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
489 |
locale submonoid = |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
490 |
fixes H and G (structure) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
491 |
assumes subset: "H \<subseteq> carrier G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
492 |
and m_closed [intro, simp]: "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
493 |
and one_closed [simp]: "\<one> \<in> H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
494 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
495 |
lemma (in submonoid) is_submonoid: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
496 |
"submonoid H G" by (rule submonoid_axioms) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
497 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
498 |
lemma (in submonoid) mem_carrier [simp]: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
499 |
"x \<in> H \<Longrightarrow> x \<in> carrier G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
500 |
using subset by blast |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
501 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
502 |
lemma (in submonoid) submonoid_is_monoid [intro]: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
503 |
assumes "monoid G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
504 |
shows "monoid (G\<lparr>carrier := H\<rparr>)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
505 |
proof - |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
506 |
interpret monoid G by fact |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
507 |
show ?thesis |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
508 |
by (simp add: monoid_def m_assoc) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
509 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
510 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
511 |
lemma submonoid_nonempty: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
512 |
"~ submonoid {} G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
513 |
by (blast dest: submonoid.one_closed) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
514 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
515 |
lemma (in submonoid) finite_monoid_imp_card_positive: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
516 |
"finite (carrier G) ==> 0 < card H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
517 |
proof (rule classical) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
518 |
assume "finite (carrier G)" and a: "~ 0 < card H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
519 |
then have "finite H" by (blast intro: finite_subset [OF subset]) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
520 |
with is_submonoid a have "submonoid {} G" by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
521 |
with submonoid_nonempty show ?thesis by contradiction |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
522 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
523 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
524 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
525 |
lemma (in monoid) monoid_incl_imp_submonoid : |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
526 |
assumes "H \<subseteq> carrier G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
527 |
and "monoid (G\<lparr>carrier := H\<rparr>)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
528 |
shows "submonoid H G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
529 |
proof (intro submonoid.intro[OF assms(1)]) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
530 |
have ab_eq : "\<And> a b. a \<in> H \<Longrightarrow> b \<in> H \<Longrightarrow> a \<otimes>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> b = a \<otimes> b" using assms by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
531 |
have "\<And>a b. a \<in> H \<Longrightarrow> b \<in> H \<Longrightarrow> a \<otimes> b \<in> carrier (G\<lparr>carrier := H\<rparr>) " |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
532 |
using assms ab_eq unfolding group_def using monoid.m_closed by fastforce |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
533 |
thus "\<And>a b. a \<in> H \<Longrightarrow> b \<in> H \<Longrightarrow> a \<otimes> b \<in> H" by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
534 |
show "\<one> \<in> H " using monoid.one_closed[OF assms(2)] assms by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
535 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
536 |
|
68517 | 537 |
lemma (in monoid) inv_unique': |
538 |
assumes "x \<in> carrier G" "y \<in> carrier G" |
|
539 |
shows "\<lbrakk> x \<otimes> y = \<one>; y \<otimes> x = \<one> \<rbrakk> \<Longrightarrow> y = inv x" |
|
540 |
proof - |
|
541 |
assume "x \<otimes> y = \<one>" and l_inv: "y \<otimes> x = \<one>" |
|
542 |
hence unit: "x \<in> Units G" |
|
543 |
using assms unfolding Units_def by auto |
|
544 |
show "y = inv x" |
|
545 |
using inv_unique[OF l_inv Units_r_inv[OF unit] assms Units_inv_closed[OF unit]] . |
|
546 |
qed |
|
547 |
||
548 |
lemma (in monoid) m_inv_monoid_consistent: (* contributed by Paulo *) |
|
549 |
assumes "x \<in> Units (G \<lparr> carrier := H \<rparr>)" and "submonoid H G" |
|
550 |
shows "inv\<^bsub>(G \<lparr> carrier := H \<rparr>)\<^esub> x = inv x" |
|
551 |
proof - |
|
552 |
have monoid: "monoid (G \<lparr> carrier := H \<rparr>)" |
|
553 |
using submonoid.submonoid_is_monoid[OF assms(2) monoid_axioms] . |
|
554 |
obtain y where y: "y \<in> H" "x \<otimes> y = \<one>" "y \<otimes> x = \<one>" |
|
555 |
using assms(1) unfolding Units_def by auto |
|
556 |
have x: "x \<in> H" and in_carrier: "x \<in> carrier G" "y \<in> carrier G" |
|
557 |
using y(1) submonoid.subset[OF assms(2)] assms(1) unfolding Units_def by auto |
|
558 |
show ?thesis |
|
559 |
using monoid.inv_unique'[OF monoid, of x y] x y |
|
560 |
using inv_unique'[OF in_carrier y(2-3)] by auto |
|
561 |
qed |
|
562 |
||
61382 | 563 |
subsection \<open>Subgroups\<close> |
13813 | 564 |
|
19783 | 565 |
locale subgroup = |
566 |
fixes H and G (structure) |
|
14963 | 567 |
assumes subset: "H \<subseteq> carrier G" |
568 |
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
|
569 |
and one_closed [simp]: "\<one> \<in> H" |
14963 | 570 |
and m_inv_closed [intro,simp]: "x \<in> H \<Longrightarrow> inv x \<in> H" |
13813 | 571 |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
572 |
lemma (in subgroup) is_subgroup: |
26199 | 573 |
"subgroup H G" by (rule subgroup_axioms) |
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
574 |
|
13813 | 575 |
declare (in subgroup) group.intro [intro] |
13949
0ce528cd6f19
HOL-Algebra complete for release Isabelle2003 (modulo section headers).
ballarin
parents:
13944
diff
changeset
|
576 |
|
14963 | 577 |
lemma (in subgroup) mem_carrier [simp]: |
578 |
"x \<in> H \<Longrightarrow> x \<in> carrier G" |
|
579 |
using subset by blast |
|
13813 | 580 |
|
14963 | 581 |
lemma (in subgroup) subgroup_is_group [intro]: |
27611 | 582 |
assumes "group G" |
583 |
shows "group (G\<lparr>carrier := H\<rparr>)" |
|
584 |
proof - |
|
29237 | 585 |
interpret group G by fact |
68458 | 586 |
have "Group.monoid (G\<lparr>carrier := H\<rparr>)" |
587 |
by (simp add: monoid_axioms submonoid.intro submonoid.submonoid_is_monoid subset) |
|
588 |
then show ?thesis |
|
589 |
by (rule monoid.group_l_invI) (auto intro: l_inv mem_carrier) |
|
27611 | 590 |
qed |
13813 | 591 |
|
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
592 |
lemma subgroup_is_submonoid: |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
593 |
assumes "subgroup H G" shows "submonoid H G" |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
594 |
using assms by (auto intro: submonoid.intro simp add: subgroup_def) |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
595 |
|
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
596 |
lemma (in group) subgroup_Units: |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
597 |
assumes "subgroup H G" shows "H \<subseteq> Units (G \<lparr> carrier := H \<rparr>)" |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
598 |
using group.Units[OF subgroup.subgroup_is_group[OF assms group_axioms]] by simp |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
599 |
|
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
600 |
lemma (in group) m_inv_consistent: |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
601 |
assumes "subgroup H G" "x \<in> H" |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
602 |
shows "inv\<^bsub>(G \<lparr> carrier := H \<rparr>)\<^esub> x = inv x" |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
603 |
using assms m_inv_monoid_consistent[OF _ subgroup_is_submonoid] subgroup_Units[of H] by auto |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
604 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
605 |
lemma (in group) int_pow_consistent: (* by Paulo *) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
606 |
assumes "subgroup H G" "x \<in> H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
607 |
shows "x [^] (n :: int) = x [^]\<^bsub>(G \<lparr> carrier := H \<rparr>)\<^esub> n" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
608 |
proof (cases) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
609 |
assume ge: "n \<ge> 0" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
610 |
hence "x [^] n = x [^] (nat n)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
611 |
using int_pow_def2 by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
612 |
also have " ... = x [^]\<^bsub>(G \<lparr> carrier := H \<rparr>)\<^esub> (nat n)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
613 |
using nat_pow_consistent by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
614 |
also have " ... = x [^]\<^bsub>(G \<lparr> carrier := H \<rparr>)\<^esub> n" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
615 |
using group.int_pow_def2[OF subgroup.subgroup_is_group[OF assms(1) is_group]] ge by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
616 |
finally show ?thesis . |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
617 |
next |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
618 |
assume "\<not> n \<ge> 0" hence lt: "n < 0" by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
619 |
hence "x [^] n = inv (x [^] (nat (- n)))" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
620 |
using int_pow_def2 by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
621 |
also have " ... = (inv x) [^] (nat (- n))" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
622 |
by (metis assms nat_pow_inv subgroup.mem_carrier) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
623 |
also have " ... = (inv\<^bsub>(G \<lparr> carrier := H \<rparr>)\<^esub> x) [^]\<^bsub>(G \<lparr> carrier := H \<rparr>)\<^esub> (nat (- n))" |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
624 |
using m_inv_consistent[OF assms] nat_pow_consistent by auto |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
625 |
also have " ... = inv\<^bsub>(G \<lparr> carrier := H \<rparr>)\<^esub> (x [^]\<^bsub>(G \<lparr> carrier := H \<rparr>)\<^esub> (nat (- n)))" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
626 |
using group.nat_pow_inv[OF subgroup.subgroup_is_group[OF assms(1) is_group]] assms(2) by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
627 |
also have " ... = x [^]\<^bsub>(G \<lparr> carrier := H \<rparr>)\<^esub> n" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
628 |
using group.int_pow_def2[OF subgroup.subgroup_is_group[OF assms(1) is_group]] lt by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
629 |
finally show ?thesis . |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
630 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
631 |
|
61382 | 632 |
text \<open> |
13813 | 633 |
Since @{term H} is nonempty, it contains some element @{term x}. Since |
63167 | 634 |
it is closed under inverse, it contains \<open>inv x\<close>. Since |
635 |
it is closed under product, it contains \<open>x \<otimes> inv x = \<one>\<close>. |
|
61382 | 636 |
\<close> |
13813 | 637 |
|
638 |
lemma (in group) one_in_subset: |
|
639 |
"[| 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 |] |
|
640 |
==> \<one> \<in> H" |
|
44472 | 641 |
by force |
13813 | 642 |
|
61382 | 643 |
text \<open>A characterization of subgroups: closed, non-empty subset.\<close> |
13813 | 644 |
|
645 |
lemma (in group) subgroupI: |
|
646 |
assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}" |
|
14963 | 647 |
and inv: "!!a. a \<in> H \<Longrightarrow> inv a \<in> H" |
648 |
and mult: "!!a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes> b \<in> H" |
|
13813 | 649 |
shows "subgroup H G" |
27714
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
650 |
proof (simp add: subgroup_def assms) |
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
651 |
show "\<one> \<in> H" by (rule one_in_subset) (auto simp only: assms) |
13813 | 652 |
qed |
653 |
||
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
654 |
lemma (in group) subgroupE: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
655 |
assumes "subgroup H G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
656 |
shows "H \<subseteq> carrier G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
657 |
and "H \<noteq> {}" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
658 |
and "\<And>a. a \<in> H \<Longrightarrow> inv a \<in> H" |
68517 | 659 |
and "\<And>a b. \<lbrakk> a \<in> H; b \<in> H \<rbrakk> \<Longrightarrow> a \<otimes> b \<in> H" |
660 |
using assms unfolding subgroup_def[of H G] by auto |
|
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
661 |
|
13936 | 662 |
declare monoid.one_closed [iff] group.inv_closed [simp] |
663 |
monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp] |
|
13813 | 664 |
|
665 |
lemma subgroup_nonempty: |
|
67091 | 666 |
"\<not> subgroup {} G" |
13813 | 667 |
by (blast dest: subgroup.one_closed) |
668 |
||
68517 | 669 |
lemma (in subgroup) finite_imp_card_positive: "finite (carrier G) \<Longrightarrow> 0 < card H" |
670 |
using subset one_closed card_gt_0_iff finite_subset by blast |
|
13813 | 671 |
|
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
672 |
(*Following 3 lemmas contributed by Martin Baillon*) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
673 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
674 |
lemma (in subgroup) subgroup_is_submonoid : |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
675 |
"submonoid H G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
676 |
by (simp add: submonoid.intro subset) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
677 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
678 |
lemma (in group) submonoid_subgroupI : |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
679 |
assumes "submonoid H G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
680 |
and "\<And>a. a \<in> H \<Longrightarrow> inv a \<in> H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
681 |
shows "subgroup H G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
682 |
by (metis assms subgroup_def submonoid_def) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
683 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
684 |
lemma (in group) group_incl_imp_subgroup: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
685 |
assumes "H \<subseteq> carrier G" |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
686 |
and "group (G\<lparr>carrier := H\<rparr>)" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
687 |
shows "subgroup H G" |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
688 |
proof (intro submonoid_subgroupI[OF monoid_incl_imp_submonoid[OF assms(1)]]) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
689 |
show "monoid (G\<lparr>carrier := H\<rparr>)" using group_def assms by blast |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
690 |
have ab_eq : "\<And> a b. a \<in> H \<Longrightarrow> b \<in> H \<Longrightarrow> a \<otimes>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> b = a \<otimes> b" using assms by simp |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
691 |
fix a assume aH : "a \<in> H" |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
692 |
have " inv\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> a \<in> carrier G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
693 |
using assms aH group.inv_closed[OF assms(2)] by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
694 |
moreover have "\<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> = \<one>" using assms monoid.one_closed ab_eq one_def by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
695 |
hence "a \<otimes>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> inv\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> a= \<one>" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
696 |
using assms ab_eq aH group.r_inv[OF assms(2)] by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
697 |
hence "a \<otimes> inv\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> a= \<one>" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
698 |
using aH assms group.inv_closed[OF assms(2)] ab_eq by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
699 |
ultimately have "inv\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> a = inv a" |
68605 | 700 |
by (metis aH assms(1) contra_subsetD group.inv_inv is_group local.inv_equality) |
701 |
moreover have "inv\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> a \<in> H" |
|
702 |
using aH group.inv_closed[OF assms(2)] by auto |
|
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
703 |
ultimately show "inv a \<in> H" by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
704 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
705 |
|
13936 | 706 |
|
61382 | 707 |
subsection \<open>Direct Products\<close> |
13813 | 708 |
|
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
709 |
definition |
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
710 |
DirProd :: "_ \<Rightarrow> _ \<Rightarrow> ('a \<times> 'b) monoid" (infixr "\<times>\<times>" 80) where |
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
711 |
"G \<times>\<times> H = |
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
712 |
\<lparr>carrier = carrier G \<times> carrier H, |
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
713 |
mult = (\<lambda>(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')), |
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
714 |
one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>" |
13813 | 715 |
|
14963 | 716 |
lemma DirProd_monoid: |
27611 | 717 |
assumes "monoid G" and "monoid H" |
14963 | 718 |
shows "monoid (G \<times>\<times> H)" |
719 |
proof - |
|
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
29240
diff
changeset
|
720 |
interpret G: monoid G by fact |
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
29240
diff
changeset
|
721 |
interpret H: monoid H by fact |
27714
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
722 |
from assms |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
723 |
show ?thesis by (unfold monoid_def DirProd_def, auto) |
14963 | 724 |
qed |
13813 | 725 |
|
726 |
||
61382 | 727 |
text\<open>Does not use the previous result because it's easier just to use auto.\<close> |
14963 | 728 |
lemma DirProd_group: |
27611 | 729 |
assumes "group G" and "group H" |
14963 | 730 |
shows "group (G \<times>\<times> H)" |
27611 | 731 |
proof - |
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
29240
diff
changeset
|
732 |
interpret G: group G by fact |
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
29240
diff
changeset
|
733 |
interpret H: group H by fact |
27611 | 734 |
show ?thesis by (rule groupI) |
14963 | 735 |
(auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv |
736 |
simp add: DirProd_def) |
|
27611 | 737 |
qed |
13813 | 738 |
|
14963 | 739 |
lemma carrier_DirProd [simp]: |
740 |
"carrier (G \<times>\<times> H) = carrier G \<times> carrier H" |
|
741 |
by (simp add: DirProd_def) |
|
13944 | 742 |
|
14963 | 743 |
lemma one_DirProd [simp]: |
744 |
"\<one>\<^bsub>G \<times>\<times> H\<^esub> = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)" |
|
745 |
by (simp add: DirProd_def) |
|
13944 | 746 |
|
14963 | 747 |
lemma mult_DirProd [simp]: |
748 |
"(g, h) \<otimes>\<^bsub>(G \<times>\<times> H)\<^esub> (g', h') = (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')" |
|
749 |
by (simp add: DirProd_def) |
|
13944 | 750 |
|
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
751 |
lemma DirProd_assoc : |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
752 |
"(G \<times>\<times> H \<times>\<times> I) = (G \<times>\<times> (H \<times>\<times> I))" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
753 |
by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
754 |
|
14963 | 755 |
lemma inv_DirProd [simp]: |
27611 | 756 |
assumes "group G" and "group H" |
13944 | 757 |
assumes g: "g \<in> carrier G" |
758 |
and h: "h \<in> carrier H" |
|
14963 | 759 |
shows "m_inv (G \<times>\<times> H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)" |
27611 | 760 |
proof - |
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
29240
diff
changeset
|
761 |
interpret G: group G by fact |
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
29240
diff
changeset
|
762 |
interpret H: group H by fact |
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
29240
diff
changeset
|
763 |
interpret Prod: group "G \<times>\<times> H" |
27714
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
764 |
by (auto intro: DirProd_group group.intro group.axioms assms) |
14963 | 765 |
show ?thesis by (simp add: Prod.inv_equality g h) |
766 |
qed |
|
27698 | 767 |
|
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
768 |
lemma DirProd_subgroups : |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
769 |
assumes "group G" |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
770 |
and "subgroup H G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
771 |
and "group K" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
772 |
and "subgroup I K" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
773 |
shows "subgroup (H \<times> I) (G \<times>\<times> K)" |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
774 |
proof (intro group.group_incl_imp_subgroup[OF DirProd_group[OF assms(1)assms(3)]]) |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
775 |
have "H \<subseteq> carrier G" "I \<subseteq> carrier K" using subgroup.subset assms apply blast+. |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
776 |
thus "(H \<times> I) \<subseteq> carrier (G \<times>\<times> K)" unfolding DirProd_def by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
777 |
have "Group.group ((G\<lparr>carrier := H\<rparr>) \<times>\<times> (K\<lparr>carrier := I\<rparr>))" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
778 |
using DirProd_group[OF subgroup.subgroup_is_group[OF assms(2)assms(1)] |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
779 |
subgroup.subgroup_is_group[OF assms(4)assms(3)]]. |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
780 |
moreover have "((G\<lparr>carrier := H\<rparr>) \<times>\<times> (K\<lparr>carrier := I\<rparr>)) = ((G \<times>\<times> K)\<lparr>carrier := H \<times> I\<rparr>)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
781 |
unfolding DirProd_def using assms apply simp. |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
782 |
ultimately show "Group.group ((G \<times>\<times> K)\<lparr>carrier := H \<times> I\<rparr>)" by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
783 |
qed |
14963 | 784 |
|
61382 | 785 |
subsection \<open>Homomorphisms and Isomorphisms\<close> |
13813 | 786 |
|
35847 | 787 |
definition |
788 |
hom :: "_ => _ => ('a => 'b) set" where |
|
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
789 |
"hom G H = |
67091 | 790 |
{h. h \<in> carrier G \<rightarrow> carrier H \<and> |
14693 | 791 |
(\<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 | 792 |
|
14761 | 793 |
lemma (in group) hom_compose: |
31754 | 794 |
"[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I" |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44655
diff
changeset
|
795 |
by (fastforce simp add: hom_def compose_def) |
13943 | 796 |
|
35848
5443079512ea
slightly more uniform definitions -- eliminated old-style meta-equality;
wenzelm
parents:
35847
diff
changeset
|
797 |
definition |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
798 |
iso :: "_ => _ => ('a => 'b) set" |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
799 |
where "iso G H = {h. h \<in> hom G H \<and> bij_betw h (carrier G) (carrier H)}" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
800 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
801 |
definition |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
802 |
is_iso :: "_ \<Rightarrow> _ \<Rightarrow> bool" (infixr "\<cong>" 60) |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
803 |
where "G \<cong> H = (iso G H \<noteq> {})" |
14761 | 804 |
|
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
805 |
lemma iso_set_refl: "(\<lambda>x. x) \<in> iso G G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
806 |
by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) |
14761 | 807 |
|
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
808 |
corollary iso_refl : "G \<cong> G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
809 |
using iso_set_refl unfolding is_iso_def by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
810 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
811 |
lemma (in group) iso_set_sym: |
68458 | 812 |
assumes "h \<in> iso G H" |
813 |
shows "inv_into (carrier G) h \<in> iso H G" |
|
814 |
proof - |
|
815 |
have h: "h \<in> hom G H" "bij_betw h (carrier G) (carrier H)" |
|
816 |
using assms by (auto simp add: iso_def bij_betw_inv_into) |
|
817 |
then have HG: "bij_betw (inv_into (carrier G) h) (carrier H) (carrier G)" |
|
818 |
by (simp add: bij_betw_inv_into) |
|
819 |
have "inv_into (carrier G) h \<in> hom H G" |
|
820 |
unfolding hom_def |
|
821 |
proof safe |
|
822 |
show *: "\<And>x. x \<in> carrier H \<Longrightarrow> inv_into (carrier G) h x \<in> carrier G" |
|
823 |
by (meson HG bij_betwE) |
|
824 |
show "inv_into (carrier G) h (x \<otimes>\<^bsub>H\<^esub> y) = inv_into (carrier G) h x \<otimes> inv_into (carrier G) h y" |
|
825 |
if "x \<in> carrier H" "y \<in> carrier H" for x y |
|
826 |
proof (rule inv_into_f_eq) |
|
827 |
show "inj_on h (carrier G)" |
|
828 |
using bij_betw_def h(2) by blast |
|
829 |
show "inv_into (carrier G) h x \<otimes> inv_into (carrier G) h y \<in> carrier G" |
|
830 |
by (simp add: * that) |
|
831 |
show "h (inv_into (carrier G) h x \<otimes> inv_into (carrier G) h y) = x \<otimes>\<^bsub>H\<^esub> y" |
|
832 |
using h bij_betw_inv_into_right [of h] unfolding hom_def by (simp add: "*" that) |
|
833 |
qed |
|
834 |
qed |
|
835 |
then show ?thesis |
|
836 |
by (simp add: Group.iso_def bij_betw_inv_into h) |
|
837 |
qed |
|
14761 | 838 |
|
68458 | 839 |
|
840 |
corollary (in group) iso_sym: "G \<cong> H \<Longrightarrow> H \<cong> G" |
|
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
841 |
using iso_set_sym unfolding is_iso_def by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
842 |
|
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
843 |
lemma (in group) iso_set_trans: |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
844 |
"[|h \<in> iso G H; i \<in> iso H I|] ==> (compose (carrier G) i h) \<in> iso G I" |
14761 | 845 |
by (auto simp add: iso_def hom_compose bij_betw_compose) |
846 |
||
68458 | 847 |
corollary (in group) iso_trans: "\<lbrakk>G \<cong> H ; H \<cong> I\<rbrakk> \<Longrightarrow> G \<cong> I" |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
848 |
using iso_set_trans unfolding is_iso_def by blast |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
849 |
|
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
850 |
(* Next four lemmas contributed by Paulo. *) |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
851 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
852 |
lemma (in monoid) hom_imp_img_monoid: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
853 |
assumes "h \<in> hom G H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
854 |
shows "monoid (H \<lparr> carrier := h ` (carrier G), one := h \<one>\<^bsub>G\<^esub> \<rparr>)" (is "monoid ?h_img") |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
855 |
proof (rule monoidI) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
856 |
show "\<one>\<^bsub>?h_img\<^esub> \<in> carrier ?h_img" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
857 |
by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
858 |
next |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
859 |
fix x y z assume "x \<in> carrier ?h_img" "y \<in> carrier ?h_img" "z \<in> carrier ?h_img" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
860 |
then obtain g1 g2 g3 |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
861 |
where g1: "g1 \<in> carrier G" "x = h g1" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
862 |
and g2: "g2 \<in> carrier G" "y = h g2" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
863 |
and g3: "g3 \<in> carrier G" "z = h g3" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
864 |
using image_iff[where ?f = h and ?A = "carrier G"] by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
865 |
have aux_lemma: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
866 |
"\<And>a b. \<lbrakk> a \<in> carrier G; b \<in> carrier G \<rbrakk> \<Longrightarrow> h a \<otimes>\<^bsub>(?h_img)\<^esub> h b = h (a \<otimes> b)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
867 |
using assms unfolding hom_def by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
868 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
869 |
show "x \<otimes>\<^bsub>(?h_img)\<^esub> \<one>\<^bsub>(?h_img)\<^esub> = x" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
870 |
using aux_lemma[OF g1(1) one_closed] g1(2) r_one[OF g1(1)] by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
871 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
872 |
show "\<one>\<^bsub>(?h_img)\<^esub> \<otimes>\<^bsub>(?h_img)\<^esub> x = x" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
873 |
using aux_lemma[OF one_closed g1(1)] g1(2) l_one[OF g1(1)] by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
874 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
875 |
have "x \<otimes>\<^bsub>(?h_img)\<^esub> y = h (g1 \<otimes> g2)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
876 |
using aux_lemma g1 g2 by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
877 |
thus "x \<otimes>\<^bsub>(?h_img)\<^esub> y \<in> carrier ?h_img" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
878 |
using g1(1) g2(1) by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
879 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
880 |
have "(x \<otimes>\<^bsub>(?h_img)\<^esub> y) \<otimes>\<^bsub>(?h_img)\<^esub> z = h ((g1 \<otimes> g2) \<otimes> g3)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
881 |
using aux_lemma g1 g2 g3 by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
882 |
also have " ... = h (g1 \<otimes> (g2 \<otimes> g3))" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
883 |
using m_assoc[OF g1(1) g2(1) g3(1)] by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
884 |
also have " ... = x \<otimes>\<^bsub>(?h_img)\<^esub> (y \<otimes>\<^bsub>(?h_img)\<^esub> z)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
885 |
using aux_lemma g1 g2 g3 by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
886 |
finally show "(x \<otimes>\<^bsub>(?h_img)\<^esub> y) \<otimes>\<^bsub>(?h_img)\<^esub> z = x \<otimes>\<^bsub>(?h_img)\<^esub> (y \<otimes>\<^bsub>(?h_img)\<^esub> z)" . |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
887 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
888 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
889 |
lemma (in group) hom_imp_img_group: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
890 |
assumes "h \<in> hom G H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
891 |
shows "group (H \<lparr> carrier := h ` (carrier G), one := h \<one>\<^bsub>G\<^esub> \<rparr>)" (is "group ?h_img") |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
892 |
proof - |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
893 |
interpret monoid ?h_img |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
894 |
using hom_imp_img_monoid[OF assms] . |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
895 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
896 |
show ?thesis |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
897 |
proof (unfold_locales) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
898 |
show "carrier ?h_img \<subseteq> Units ?h_img" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
899 |
proof (auto simp add: Units_def) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
900 |
have aux_lemma: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
901 |
"\<And>g1 g2. \<lbrakk> g1 \<in> carrier G; g2 \<in> carrier G \<rbrakk> \<Longrightarrow> h g1 \<otimes>\<^bsub>H\<^esub> h g2 = h (g1 \<otimes> g2)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
902 |
using assms unfolding hom_def by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
903 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
904 |
fix g1 assume g1: "g1 \<in> carrier G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
905 |
thus "\<exists>g2 \<in> carrier G. (h g2) \<otimes>\<^bsub>H\<^esub> (h g1) = h \<one> \<and> (h g1) \<otimes>\<^bsub>H\<^esub> (h g2) = h \<one>" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
906 |
using aux_lemma[OF g1 inv_closed[OF g1]] |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
907 |
aux_lemma[OF inv_closed[OF g1] g1] |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
908 |
inv_closed by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
909 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
910 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
911 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
912 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
913 |
lemma (in group) iso_imp_group: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
914 |
assumes "G \<cong> H" and "monoid H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
915 |
shows "group H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
916 |
proof - |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
917 |
obtain \<phi> where phi: "\<phi> \<in> iso G H" "inv_into (carrier G) \<phi> \<in> iso H G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
918 |
using iso_set_sym assms unfolding is_iso_def by blast |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
919 |
define \<psi> where psi_def: "\<psi> = inv_into (carrier G) \<phi>" |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
920 |
|
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
921 |
from phi |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
922 |
have surj: "\<phi> ` (carrier G) = (carrier H)" "\<psi> ` (carrier H) = (carrier G)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
923 |
and inj: "inj_on \<phi> (carrier G)" "inj_on \<psi> (carrier H)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
924 |
and phi_hom: "\<And>g1 g2. \<lbrakk> g1 \<in> carrier G; g2 \<in> carrier G \<rbrakk> \<Longrightarrow> \<phi> (g1 \<otimes> g2) = (\<phi> g1) \<otimes>\<^bsub>H\<^esub> (\<phi> g2)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
925 |
and psi_hom: "\<And>h1 h2. \<lbrakk> h1 \<in> carrier H; h2 \<in> carrier H \<rbrakk> \<Longrightarrow> \<psi> (h1 \<otimes>\<^bsub>H\<^esub> h2) = (\<psi> h1) \<otimes> (\<psi> h2)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
926 |
using psi_def unfolding iso_def bij_betw_def hom_def by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
927 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
928 |
have phi_one: "\<phi> \<one> = \<one>\<^bsub>H\<^esub>" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
929 |
proof - |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
930 |
have "(\<phi> \<one>) \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = (\<phi> \<one>) \<otimes>\<^bsub>H\<^esub> (\<phi> \<one>)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
931 |
by (metis assms(2) image_eqI monoid.r_one one_closed phi_hom r_one surj(1)) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
932 |
thus ?thesis |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
933 |
by (metis (no_types, hide_lams) Units_eq Units_one_closed assms(2) f_inv_into_f imageI |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
934 |
monoid.l_one monoid.one_closed phi_hom psi_def r_one surj) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
935 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
936 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
937 |
have "carrier H \<subseteq> Units H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
938 |
proof |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
939 |
fix h assume h: "h \<in> carrier H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
940 |
let ?inv_h = "\<phi> (inv (\<psi> h))" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
941 |
have "h \<otimes>\<^bsub>H\<^esub> ?inv_h = \<phi> (\<psi> h) \<otimes>\<^bsub>H\<^esub> ?inv_h" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
942 |
by (simp add: f_inv_into_f h psi_def surj(1)) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
943 |
also have " ... = \<phi> ((\<psi> h) \<otimes> inv (\<psi> h))" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
944 |
by (metis h imageI inv_closed phi_hom surj(2)) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
945 |
also have " ... = \<phi> \<one>" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
946 |
by (simp add: h inv_into_into psi_def surj(1)) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
947 |
finally have 1: "h \<otimes>\<^bsub>H\<^esub> ?inv_h = \<one>\<^bsub>H\<^esub>" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
948 |
using phi_one by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
949 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
950 |
have "?inv_h \<otimes>\<^bsub>H\<^esub> h = ?inv_h \<otimes>\<^bsub>H\<^esub> \<phi> (\<psi> h)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
951 |
by (simp add: f_inv_into_f h psi_def surj(1)) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
952 |
also have " ... = \<phi> (inv (\<psi> h) \<otimes> (\<psi> h))" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
953 |
by (metis h imageI inv_closed phi_hom surj(2)) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
954 |
also have " ... = \<phi> \<one>" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
955 |
by (simp add: h inv_into_into psi_def surj(1)) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
956 |
finally have 2: "?inv_h \<otimes>\<^bsub>H\<^esub> h = \<one>\<^bsub>H\<^esub>" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
957 |
using phi_one by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
958 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
959 |
thus "h \<in> Units H" unfolding Units_def using 1 2 h surj by fastforce |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
960 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
961 |
thus ?thesis unfolding group_def group_axioms_def using assms(2) by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
962 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
963 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
964 |
corollary (in group) iso_imp_img_group: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
965 |
assumes "h \<in> iso G H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
966 |
shows "group (H \<lparr> one := h \<one> \<rparr>)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
967 |
proof - |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
968 |
let ?h_img = "H \<lparr> carrier := h ` (carrier G), one := h \<one> \<rparr>" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
969 |
have "h \<in> iso G ?h_img" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
970 |
using assms unfolding iso_def hom_def bij_betw_def by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
971 |
hence "G \<cong> ?h_img" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
972 |
unfolding is_iso_def by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
973 |
hence "group ?h_img" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
974 |
using iso_imp_group[of ?h_img] hom_imp_img_monoid[of h H] assms unfolding iso_def by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
975 |
moreover have "carrier H = carrier ?h_img" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
976 |
using assms unfolding iso_def bij_betw_def by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
977 |
hence "H \<lparr> one := h \<one> \<rparr> = ?h_img" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
978 |
by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
979 |
ultimately show ?thesis by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
980 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
981 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
982 |
lemma DirProd_commute_iso_set: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
983 |
shows "(\<lambda>(x,y). (y,x)) \<in> iso (G \<times>\<times> H) (H \<times>\<times> G)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
984 |
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
985 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
986 |
corollary DirProd_commute_iso : |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
987 |
"(G \<times>\<times> H) \<cong> (H \<times>\<times> G)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
988 |
using DirProd_commute_iso_set unfolding is_iso_def by blast |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
989 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
990 |
lemma DirProd_assoc_iso_set: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
991 |
shows "(\<lambda>(x,y,z). (x,(y,z))) \<in> iso (G \<times>\<times> H \<times>\<times> I) (G \<times>\<times> (H \<times>\<times> I))" |
31754 | 992 |
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def) |
14761 | 993 |
|
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
994 |
lemma (in group) DirProd_iso_set_trans: |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
995 |
assumes "g \<in> iso G G2" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
996 |
and "h \<in> iso H I" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
997 |
shows "(\<lambda>(x,y). (g x, h y)) \<in> iso (G \<times>\<times> H) (G2 \<times>\<times> I)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
998 |
proof- |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
999 |
have "(\<lambda>(x,y). (g x, h y)) \<in> hom (G \<times>\<times> H) (G2 \<times>\<times> I)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1000 |
using assms unfolding iso_def hom_def by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1001 |
moreover have " inj_on (\<lambda>(x,y). (g x, h y)) (carrier (G \<times>\<times> H))" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1002 |
using assms unfolding iso_def DirProd_def bij_betw_def inj_on_def by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1003 |
moreover have "(\<lambda>(x, y). (g x, h y)) ` carrier (G \<times>\<times> H) = carrier (G2 \<times>\<times> I)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1004 |
using assms unfolding iso_def bij_betw_def image_def DirProd_def by fastforce |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1005 |
ultimately show "(\<lambda>(x,y). (g x, h y)) \<in> iso (G \<times>\<times> H) (G2 \<times>\<times> I)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1006 |
unfolding iso_def bij_betw_def by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1007 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1008 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1009 |
corollary (in group) DirProd_iso_trans : |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1010 |
assumes "G \<cong> G2" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1011 |
and "H \<cong> I" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1012 |
shows "G \<times>\<times> H \<cong> G2 \<times>\<times> I" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1013 |
using DirProd_iso_set_trans assms unfolding is_iso_def by blast |
14761 | 1014 |
|
1015 |
||
61382 | 1016 |
text\<open>Basis for homomorphism proofs: we assume two groups @{term G} and |
1017 |
@{term H}, with a homomorphism @{term h} between them\<close> |
|
61565
352c73a689da
Qualifiers in locale expressions default to mandatory regardless of the command.
ballarin
parents:
61384
diff
changeset
|
1018 |
locale group_hom = G?: group G + H?: group H for G (structure) and H (structure) + |
29237 | 1019 |
fixes h |
13813 | 1020 |
assumes homh: "h \<in> hom G H" |
29240 | 1021 |
|
1022 |
lemma (in group_hom) hom_mult [simp]: |
|
1023 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y" |
|
1024 |
proof - |
|
1025 |
assume "x \<in> carrier G" "y \<in> carrier G" |
|
1026 |
with homh [unfolded hom_def] show ?thesis by simp |
|
1027 |
qed |
|
1028 |
||
1029 |
lemma (in group_hom) hom_closed [simp]: |
|
1030 |
"x \<in> carrier G ==> h x \<in> carrier H" |
|
1031 |
proof - |
|
1032 |
assume "x \<in> carrier G" |
|
31754 | 1033 |
with homh [unfolded hom_def] show ?thesis by auto |
29240 | 1034 |
qed |
13813 | 1035 |
|
1036 |
lemma (in group_hom) one_closed [simp]: |
|
1037 |
"h \<one> \<in> carrier H" |
|
1038 |
by simp |
|
1039 |
||
1040 |
lemma (in group_hom) hom_one [simp]: |
|
14693 | 1041 |
"h \<one> = \<one>\<^bsub>H\<^esub>" |
13813 | 1042 |
proof - |
15076
4b3d280ef06a
New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents:
14963
diff
changeset
|
1043 |
have "h \<one> \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = h \<one> \<otimes>\<^bsub>H\<^esub> h \<one>" |
13813 | 1044 |
by (simp add: hom_mult [symmetric] del: hom_mult) |
1045 |
then show ?thesis by (simp del: r_one) |
|
1046 |
qed |
|
1047 |
||
1048 |
lemma (in group_hom) inv_closed [simp]: |
|
1049 |
"x \<in> carrier G ==> h (inv x) \<in> carrier H" |
|
1050 |
by simp |
|
1051 |
||
1052 |
lemma (in group_hom) hom_inv [simp]: |
|
14693 | 1053 |
"x \<in> carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)" |
13813 | 1054 |
proof - |
1055 |
assume x: "x \<in> carrier G" |
|
14693 | 1056 |
then have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = \<one>\<^bsub>H\<^esub>" |
14963 | 1057 |
by (simp add: hom_mult [symmetric] del: hom_mult) |
14693 | 1058 |
also from x have "... = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" |
14963 | 1059 |
by (simp add: hom_mult [symmetric] del: hom_mult) |
14693 | 1060 |
finally have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" . |
27698 | 1061 |
with x show ?thesis by (simp del: H.r_inv H.Units_r_inv) |
13813 | 1062 |
qed |
1063 |
||
57271 | 1064 |
(* Contributed by Joachim Breitner *) |
1065 |
lemma (in group) int_pow_is_hom: |
|
67399 | 1066 |
"x \<in> carrier G \<Longrightarrow> (([^]) x) \<in> hom \<lparr> carrier = UNIV, mult = (+), one = 0::int \<rparr> G " |
57271 | 1067 |
unfolding hom_def by (simp add: int_pow_mult) |
1068 |
||
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1069 |
(* Next six lemmas contributed by Paulo. *) |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1070 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1071 |
lemma (in group_hom) img_is_subgroup: "subgroup (h ` (carrier G)) H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1072 |
apply (rule subgroupI) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1073 |
apply (auto simp add: image_subsetI) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1074 |
apply (metis (no_types, hide_lams) G.inv_closed hom_inv image_iff) |
68605 | 1075 |
by (metis G.monoid_axioms hom_mult image_eqI monoid.m_closed) |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1076 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1077 |
lemma (in group_hom) subgroup_img_is_subgroup: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1078 |
assumes "subgroup I G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1079 |
shows "subgroup (h ` I) H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1080 |
proof - |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1081 |
have "h \<in> hom (G \<lparr> carrier := I \<rparr>) H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1082 |
using G.subgroupE[OF assms] subgroup.mem_carrier[OF assms] homh |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1083 |
unfolding hom_def by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1084 |
hence "group_hom (G \<lparr> carrier := I \<rparr>) H h" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1085 |
using subgroup.subgroup_is_group[OF assms G.is_group] is_group |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1086 |
unfolding group_hom_def group_hom_axioms_def by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1087 |
thus ?thesis |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1088 |
using group_hom.img_is_subgroup[of "G \<lparr> carrier := I \<rparr>" H h] by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1089 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1090 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1091 |
lemma (in group_hom) induced_group_hom: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1092 |
assumes "subgroup I G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1093 |
shows "group_hom (G \<lparr> carrier := I \<rparr>) (H \<lparr> carrier := h ` I \<rparr>) h" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1094 |
proof - |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1095 |
have "h \<in> hom (G \<lparr> carrier := I \<rparr>) (H \<lparr> carrier := h ` I \<rparr>)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1096 |
using homh subgroup.mem_carrier[OF assms] unfolding hom_def by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1097 |
thus ?thesis |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1098 |
unfolding group_hom_def group_hom_axioms_def |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1099 |
using subgroup.subgroup_is_group[OF assms G.is_group] |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1100 |
subgroup.subgroup_is_group[OF subgroup_img_is_subgroup[OF assms] is_group] by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1101 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1102 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1103 |
lemma (in group) canonical_inj_is_hom: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1104 |
assumes "subgroup H G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1105 |
shows "group_hom (G \<lparr> carrier := H \<rparr>) G id" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1106 |
unfolding group_hom_def group_hom_axioms_def hom_def |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1107 |
using subgroup.subgroup_is_group[OF assms is_group] |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1108 |
is_group subgroup.subset[OF assms] by auto |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1109 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1110 |
lemma (in group_hom) nat_pow_hom: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1111 |
"x \<in> carrier G \<Longrightarrow> h (x [^] (n :: nat)) = (h x) [^]\<^bsub>H\<^esub> n" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1112 |
by (induction n) auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1113 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1114 |
lemma (in group_hom) int_pow_hom: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1115 |
"x \<in> carrier G \<Longrightarrow> h (x [^] (n :: int)) = (h x) [^]\<^bsub>H\<^esub> n" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1116 |
using int_pow_def2 nat_pow_hom by (simp add: G.int_pow_def2) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1117 |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
1118 |
|
61382 | 1119 |
subsection \<open>Commutative Structures\<close> |
13936 | 1120 |
|
61382 | 1121 |
text \<open> |
13936 | 1122 |
Naming convention: multiplicative structures that are commutative |
1123 |
are called \emph{commutative}, additive structures are called |
|
1124 |
\emph{Abelian}. |
|
61382 | 1125 |
\<close> |
13813 | 1126 |
|
14963 | 1127 |
locale comm_monoid = monoid + |
1128 |
assumes m_comm: "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x" |
|
13813 | 1129 |
|
14963 | 1130 |
lemma (in comm_monoid) m_lcomm: |
1131 |
"\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> |
|
13813 | 1132 |
x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)" |
1133 |
proof - |
|
14693 | 1134 |
assume xyz: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
13813 | 1135 |
from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc) |
1136 |
also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm) |
|
1137 |
also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc) |
|
1138 |
finally show ?thesis . |
|
1139 |
qed |
|
1140 |
||
14963 | 1141 |
lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm |
13813 | 1142 |
|
13936 | 1143 |
lemma comm_monoidI: |
19783 | 1144 |
fixes G (structure) |
13936 | 1145 |
assumes m_closed: |
14693 | 1146 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G" |
1147 |
and one_closed: "\<one> \<in> carrier G" |
|
13936 | 1148 |
and m_assoc: |
1149 |
"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
14693 | 1150 |
(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
1151 |
and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" |
|
13936 | 1152 |
and m_comm: |
14693 | 1153 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x" |
13936 | 1154 |
shows "comm_monoid G" |
1155 |
using l_one |
|
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1156 |
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
|
1157 |
intro: assms simp: m_closed one_closed m_comm) |
13817 | 1158 |
|
13936 | 1159 |
lemma (in monoid) monoid_comm_monoidI: |
1160 |
assumes m_comm: |
|
14693 | 1161 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x" |
13936 | 1162 |
shows "comm_monoid G" |
1163 |
by (rule comm_monoidI) (auto intro: m_assoc m_comm) |
|
14963 | 1164 |
|
13936 | 1165 |
lemma (in comm_monoid) nat_pow_distr: |
1166 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> |
|
67341
df79ef3b3a41
Renamed (^) to [^] in preparation of the move from "op X" to (X)
nipkow
parents:
67091
diff
changeset
|
1167 |
(x \<otimes> y) [^] (n::nat) = x [^] n \<otimes> y [^] n" |
13936 | 1168 |
by (induct n) (simp, simp add: m_ac) |
1169 |
||
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1170 |
lemma (in comm_monoid) submonoid_is_comm_monoid : |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1171 |
assumes "submonoid H G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1172 |
shows "comm_monoid (G\<lparr>carrier := H\<rparr>)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1173 |
proof (intro monoid.monoid_comm_monoidI) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1174 |
show "monoid (G\<lparr>carrier := H\<rparr>)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1175 |
using submonoid.submonoid_is_monoid assms comm_monoid_axioms comm_monoid_def by blast |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1176 |
show "\<And>x y. x \<in> carrier (G\<lparr>carrier := H\<rparr>) \<Longrightarrow> y \<in> carrier (G\<lparr>carrier := H\<rparr>) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1177 |
\<Longrightarrow> x \<otimes>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> y = y \<otimes>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> x" apply simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1178 |
using assms comm_monoid_axioms_def submonoid.mem_carrier |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1179 |
by (metis m_comm) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1180 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1181 |
|
13936 | 1182 |
locale comm_group = comm_monoid + group |
1183 |
||
1184 |
lemma (in group) group_comm_groupI: |
|
1185 |
assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> |
|
14693 | 1186 |
x \<otimes> y = y \<otimes> x" |
13936 | 1187 |
shows "comm_group G" |
61169 | 1188 |
by standard (simp_all add: m_comm) |
13817 | 1189 |
|
13936 | 1190 |
lemma comm_groupI: |
19783 | 1191 |
fixes G (structure) |
13936 | 1192 |
assumes m_closed: |
14693 | 1193 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G" |
1194 |
and one_closed: "\<one> \<in> carrier G" |
|
13936 | 1195 |
and m_assoc: |
1196 |
"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
14693 | 1197 |
(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
13936 | 1198 |
and m_comm: |
14693 | 1199 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x" |
1200 |
and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" |
|
14963 | 1201 |
and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
13936 | 1202 |
shows "comm_group G" |
27714
27b4d7c01f8b
Tuned (for the sake of a meaningless log entry).
ballarin
parents:
27713
diff
changeset
|
1203 |
by (fast intro: group.group_comm_groupI groupI assms) |
13936 | 1204 |
|
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1205 |
lemma comm_groupE: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1206 |
fixes G (structure) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1207 |
assumes "comm_group G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1208 |
shows "\<And>x y. \<lbrakk> x \<in> carrier G; y \<in> carrier G \<rbrakk> \<Longrightarrow> x \<otimes> y \<in> carrier G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1209 |
and "\<one> \<in> carrier G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1210 |
and "\<And>x y z. \<lbrakk> x \<in> carrier G; y \<in> carrier G; z \<in> carrier G \<rbrakk> \<Longrightarrow> (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1211 |
and "\<And>x y. \<lbrakk> x \<in> carrier G; y \<in> carrier G \<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1212 |
and "\<And>x. x \<in> carrier G \<Longrightarrow> \<one> \<otimes> x = x" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1213 |
and "\<And>x. x \<in> carrier G \<Longrightarrow> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1214 |
apply (simp_all add: group.axioms assms comm_group.axioms comm_monoid.m_comm comm_monoid.m_ac(1)) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1215 |
by (simp_all add: Group.group.axioms(1) assms comm_group.axioms(2) monoid.m_closed group.r_inv_ex) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1216 |
|
13936 | 1217 |
lemma (in comm_group) inv_mult: |
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
1218 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y" |
13936 | 1219 |
by (simp add: m_ac inv_mult_group) |
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
1220 |
|
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1221 |
(* Next three lemmas contributed by Paulo. *) |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1222 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1223 |
lemma (in comm_monoid) hom_imp_img_comm_monoid: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1224 |
assumes "h \<in> hom G H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1225 |
shows "comm_monoid (H \<lparr> carrier := h ` (carrier G), one := h \<one>\<^bsub>G\<^esub> \<rparr>)" (is "comm_monoid ?h_img") |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1226 |
proof (rule monoid.monoid_comm_monoidI) |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1227 |
show "monoid ?h_img" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1228 |
using hom_imp_img_monoid[OF assms] . |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1229 |
next |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1230 |
fix x y assume "x \<in> carrier ?h_img" "y \<in> carrier ?h_img" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1231 |
then obtain g1 g2 |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1232 |
where g1: "g1 \<in> carrier G" "x = h g1" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1233 |
and g2: "g2 \<in> carrier G" "y = h g2" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1234 |
by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1235 |
have "x \<otimes>\<^bsub>(?h_img)\<^esub> y = h (g1 \<otimes> g2)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1236 |
using g1 g2 assms unfolding hom_def by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1237 |
also have " ... = h (g2 \<otimes> g1)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1238 |
using m_comm[OF g1(1) g2(1)] by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1239 |
also have " ... = y \<otimes>\<^bsub>(?h_img)\<^esub> x" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1240 |
using g1 g2 assms unfolding hom_def by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1241 |
finally show "x \<otimes>\<^bsub>(?h_img)\<^esub> y = y \<otimes>\<^bsub>(?h_img)\<^esub> x" . |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1242 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1243 |
|
68517 | 1244 |
lemma (in comm_group) hom_imp_img_comm_group: |
1245 |
assumes "h \<in> hom G H" |
|
1246 |
shows "comm_group (H \<lparr> carrier := h ` (carrier G), one := h \<one>\<^bsub>G\<^esub> \<rparr>)" |
|
1247 |
unfolding comm_group_def |
|
1248 |
using hom_imp_img_group[OF assms] hom_imp_img_comm_monoid[OF assms] by simp |
|
1249 |
||
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1250 |
lemma (in comm_group) iso_imp_img_comm_group: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1251 |
assumes "h \<in> iso G H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1252 |
shows "comm_group (H \<lparr> one := h \<one>\<^bsub>G\<^esub> \<rparr>)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1253 |
proof - |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1254 |
let ?h_img = "H \<lparr> carrier := h ` (carrier G), one := h \<one> \<rparr>" |
68517 | 1255 |
have "comm_group ?h_img" |
1256 |
using hom_imp_img_comm_group[of h H] assms unfolding iso_def by auto |
|
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1257 |
moreover have "carrier H = carrier ?h_img" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1258 |
using assms unfolding iso_def bij_betw_def by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1259 |
hence "H \<lparr> one := h \<one> \<rparr> = ?h_img" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1260 |
by simp |
68517 | 1261 |
ultimately show ?thesis by simp |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1262 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1263 |
|
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1264 |
lemma (in comm_group) iso_imp_comm_group: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1265 |
assumes "G \<cong> H" "monoid H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1266 |
shows "comm_group H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1267 |
proof - |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1268 |
obtain h where h: "h \<in> iso G H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1269 |
using assms(1) unfolding is_iso_def by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1270 |
hence comm_gr: "comm_group (H \<lparr> one := h \<one> \<rparr>)" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1271 |
using iso_imp_img_comm_group[of h H] by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1272 |
hence "\<And>x. x \<in> carrier H \<Longrightarrow> h \<one> \<otimes>\<^bsub>H\<^esub> x = x" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1273 |
using monoid.l_one[of "H \<lparr> one := h \<one> \<rparr>"] unfolding comm_group_def comm_monoid_def by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1274 |
moreover have "h \<one> \<in> carrier H" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1275 |
using h one_closed unfolding iso_def hom_def by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1276 |
ultimately have "h \<one> = \<one>\<^bsub>H\<^esub>" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1277 |
using monoid.one_unique[OF assms(2), of "h \<one>"] by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1278 |
hence "H = H \<lparr> one := h \<one> \<rparr>" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1279 |
by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1280 |
thus ?thesis |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1281 |
using comm_gr by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1282 |
qed |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1283 |
|
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1284 |
(*A subgroup of a subgroup is a subgroup of the group*) |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1285 |
lemma (in group) incl_subgroup: |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1286 |
assumes "subgroup J G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1287 |
and "subgroup I (G\<lparr>carrier:=J\<rparr>)" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1288 |
shows "subgroup I G" unfolding subgroup_def |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1289 |
proof |
68452
c027dfbfad30
more on infinite products. Also subgroup_imp_subset -> subgroup.subset
paulson <lp15@cam.ac.uk>
parents:
68445
diff
changeset
|
1290 |
have H1: "I \<subseteq> carrier (G\<lparr>carrier:=J\<rparr>)" using assms(2) subgroup.subset by blast |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1291 |
also have H2: "...\<subseteq>J" by simp |
68452
c027dfbfad30
more on infinite products. Also subgroup_imp_subset -> subgroup.subset
paulson <lp15@cam.ac.uk>
parents:
68445
diff
changeset
|
1292 |
also have "...\<subseteq>(carrier G)" by (simp add: assms(1) subgroup.subset) |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1293 |
finally have H: "I \<subseteq> carrier G" by simp |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1294 |
have "(\<And>x y. \<lbrakk>x \<in> I ; y \<in> I\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> I)" using assms(2) by (auto simp add: subgroup_def) |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1295 |
thus "I \<subseteq> carrier G \<and> (\<forall>x y. x \<in> I \<longrightarrow> y \<in> I \<longrightarrow> x \<otimes> y \<in> I)" using H by blast |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1296 |
have K: "\<one> \<in> I" using assms(2) by (auto simp add: subgroup_def) |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1297 |
have "(\<And>x. x \<in> I \<Longrightarrow> inv x \<in> I)" using assms subgroup.m_inv_closed H |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
1298 |
by (metis H1 H2 m_inv_consistent subsetCE) |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1299 |
thus "\<one> \<in> I \<and> (\<forall>x. x \<in> I \<longrightarrow> inv x \<in> I)" using K by blast |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1300 |
qed |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1301 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1302 |
(*A subgroup included in another subgroup is a subgroup of the subgroup*) |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1303 |
lemma (in group) subgroup_incl: |
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
1304 |
assumes "subgroup I G" and "subgroup J G" and "I \<subseteq> J" |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
1305 |
shows "subgroup I (G \<lparr> carrier := J \<rparr>)" |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
1306 |
using group.group_incl_imp_subgroup[of "G \<lparr> carrier := J \<rparr>" I] |
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
1307 |
assms(1-2)[THEN subgroup.subgroup_is_group[OF _ group_axioms]] assms(3) by auto |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1308 |
|
20318
0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset
|
1309 |
|
61382 | 1310 |
subsection \<open>The Lattice of Subgroups of a Group\<close> |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1311 |
|
61382 | 1312 |
text_raw \<open>\label{sec:subgroup-lattice}\<close> |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1313 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1314 |
theorem (in group) subgroups_partial_order: |
67399 | 1315 |
"partial_order \<lparr>carrier = {H. subgroup H G}, eq = (=), le = (\<subseteq>)\<rparr>" |
61169 | 1316 |
by standard simp_all |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1317 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1318 |
lemma (in group) subgroup_self: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1319 |
"subgroup (carrier G) G" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1320 |
by (rule subgroupI) auto |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1321 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1322 |
lemma (in group) subgroup_imp_group: |
55926 | 1323 |
"subgroup H G ==> group (G\<lparr>carrier := H\<rparr>)" |
26199 | 1324 |
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
|
1325 |
|
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1326 |
lemma (in group) is_monoid [intro, simp]: |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1327 |
"monoid G" |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1328 |
by (auto intro: monoid.intro m_assoc) |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1329 |
|
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1330 |
lemma (in group) subgroup_mult_equality: |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1331 |
"\<lbrakk> subgroup H G; h1 \<in> H; h2 \<in> H \<rbrakk> \<Longrightarrow> h1 \<otimes>\<^bsub>G \<lparr> carrier := H \<rparr>\<^esub> h2 = h1 \<otimes> h2" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1332 |
unfolding subgroup_def by simp |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1333 |
|
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1334 |
theorem (in group) subgroups_Inter: |
67091 | 1335 |
assumes subgr: "(\<And>H. H \<in> A \<Longrightarrow> subgroup H G)" |
1336 |
and not_empty: "A \<noteq> {}" |
|
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1337 |
shows "subgroup (\<Inter>A) G" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1338 |
proof (rule subgroupI) |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1339 |
from subgr [THEN subgroup.subset] and not_empty |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1340 |
show "\<Inter>A \<subseteq> carrier G" by blast |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1341 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1342 |
from subgr [THEN subgroup.one_closed] |
67091 | 1343 |
show "\<Inter>A \<noteq> {}" by blast |
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1344 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1345 |
fix x assume "x \<in> \<Inter>A" |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1346 |
with subgr [THEN subgroup.m_inv_closed] |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1347 |
show "inv x \<in> \<Inter>A" by blast |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1348 |
next |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1349 |
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
|
1350 |
with subgr [THEN subgroup.m_closed] |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1351 |
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
|
1352 |
qed |
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset
|
1353 |
|
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1354 |
lemma (in group) subgroups_Inter_pair : |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1355 |
assumes "subgroup I G" |
68443
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1356 |
and "subgroup J G" |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1357 |
shows "subgroup (I\<inter>J) G" using subgroups_Inter[ where ?A = "{I,J}"] assms by auto |
43055b016688
New material from Martin Baillon and Paulo EmÃlio de Vilhena
paulson <lp15@cam.ac.uk>
parents:
68399
diff
changeset
|
1358 |
|
66579 | 1359 |
theorem (in group) subgroups_complete_lattice: |
67399 | 1360 |
"complete_lattice \<lparr>carrier = {H. subgroup H G}, eq = (=), le = (\<subseteq>)\<rparr>" |
66579 | 1361 |
(is "complete_lattice ?L") |
1362 |
proof (rule partial_order.complete_lattice_criterion1) |
|
1363 |
show "partial_order ?L" by (rule subgroups_partial_order) |
|
1364 |
next |
|
1365 |
have "greatest ?L (carrier G) (carrier ?L)" |
|
1366 |
by (unfold greatest_def) (simp add: subgroup.subset subgroup_self) |
|
1367 |
then show "\<exists>G. greatest ?L G (carrier ?L)" .. |
|
1368 |
next |
|
1369 |
fix A |
|
67091 | 1370 |
assume L: "A \<subseteq> carrier ?L" and non_empty: "A \<noteq> {}" |
66579 | 1371 |
then have Int_subgroup: "subgroup (\<Inter>A) G" |
1372 |
by (fastforce intro: subgroups_Inter) |
|
1373 |
have "greatest ?L (\<Inter>A) (Lower ?L A)" (is "greatest _ ?Int _") |
|
1374 |
proof (rule greatest_LowerI) |
|
1375 |
fix H |
|
1376 |
assume H: "H \<in> A" |
|
1377 |
with L have subgroupH: "subgroup H G" by auto |
|
1378 |
from subgroupH have groupH: "group (G \<lparr>carrier := H\<rparr>)" (is "group ?H") |
|
1379 |
by (rule subgroup_imp_group) |
|
1380 |
from groupH have monoidH: "monoid ?H" |
|
1381 |
by (rule group.is_monoid) |
|
1382 |
from H have Int_subset: "?Int \<subseteq> H" by fastforce |
|
1383 |
then show "le ?L ?Int H" by simp |
|
1384 |
next |
|
1385 |
fix H |
|
1386 |
assume H: "H \<in> Lower ?L A" |
|
1387 |
with L Int_subgroup show "le ?L H ?Int" |
|
1388 |
by (fastforce simp: Lower_def intro: Inter_greatest) |
|
1389 |
next |
|
1390 |
show "A \<subseteq> carrier ?L" by (rule L) |
|
1391 |
next |
|
1392 |
show "?Int \<in> carrier ?L" by simp (rule Int_subgroup) |
|
1393 |
qed |
|
1394 |
then show "\<exists>I. greatest ?L I (Lower ?L A)" .. |
|
1395 |
qed |
|
1396 |
||
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1397 |
subsection\<open>Jeremy Avigad's @{text"More_Group"} material\<close> |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1398 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1399 |
text \<open> |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1400 |
Show that the units in any monoid give rise to a group. |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1401 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1402 |
The file Residues.thy provides some infrastructure to use |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1403 |
facts about the unit group within the ring locale. |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1404 |
\<close> |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1405 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1406 |
definition units_of :: "('a, 'b) monoid_scheme \<Rightarrow> 'a monoid" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1407 |
where "units_of G = |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1408 |
\<lparr>carrier = Units G, Group.monoid.mult = Group.monoid.mult G, one = one G\<rparr>" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1409 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1410 |
lemma (in monoid) units_group: "group (units_of G)" |
68458 | 1411 |
proof - |
1412 |
have "\<And>x y z. \<lbrakk>x \<in> Units G; y \<in> Units G; z \<in> Units G\<rbrakk> \<Longrightarrow> x \<otimes> y \<otimes> z = x \<otimes> (y \<otimes> z)" |
|
1413 |
by (simp add: Units_closed m_assoc) |
|
1414 |
moreover have "\<And>x. x \<in> Units G \<Longrightarrow> \<exists>y\<in>Units G. y \<otimes> x = \<one>" |
|
1415 |
using Units_l_inv by blast |
|
1416 |
ultimately show ?thesis |
|
1417 |
unfolding units_of_def |
|
1418 |
by (force intro!: groupI) |
|
1419 |
qed |
|
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1420 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1421 |
lemma (in comm_monoid) units_comm_group: "comm_group (units_of G)" |
68458 | 1422 |
proof - |
1423 |
have "\<And>x y. \<lbrakk>x \<in> carrier (units_of G); y \<in> carrier (units_of G)\<rbrakk> |
|
1424 |
\<Longrightarrow> x \<otimes>\<^bsub>units_of G\<^esub> y = y \<otimes>\<^bsub>units_of G\<^esub> x" |
|
1425 |
by (simp add: Units_closed m_comm units_of_def) |
|
1426 |
then show ?thesis |
|
1427 |
by (rule group.group_comm_groupI [OF units_group]) auto |
|
1428 |
qed |
|
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1429 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1430 |
lemma units_of_carrier: "carrier (units_of G) = Units G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1431 |
by (auto simp: units_of_def) |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1432 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1433 |
lemma units_of_mult: "mult (units_of G) = mult G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1434 |
by (auto simp: units_of_def) |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1435 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1436 |
lemma units_of_one: "one (units_of G) = one G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1437 |
by (auto simp: units_of_def) |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1438 |
|
68555
22d51874f37d
a few more lemmas from Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68551
diff
changeset
|
1439 |
lemma (in monoid) units_of_inv: |
68458 | 1440 |
assumes "x \<in> Units G" |
1441 |
shows "m_inv (units_of G) x = m_inv G x" |
|
1442 |
by (simp add: assms group.inv_equality units_group units_of_carrier units_of_mult units_of_one) |
|
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1443 |
|
68551
b680e74eb6f2
More on Algebra by Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1444 |
lemma units_of_units [simp] : "Units (units_of G) = Units G" |
b680e74eb6f2
More on Algebra by Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1445 |
unfolding units_of_def Units_def by force |
b680e74eb6f2
More on Algebra by Paulo and Martin
paulson <lp15@cam.ac.uk>
parents:
68517
diff
changeset
|
1446 |
|
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1447 |
lemma (in group) surj_const_mult: "a \<in> carrier G \<Longrightarrow> (\<lambda>x. a \<otimes> x) ` carrier G = carrier G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1448 |
apply (auto simp add: image_def) |
68458 | 1449 |
by (metis inv_closed inv_solve_left m_closed) |
68445
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1450 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1451 |
lemma (in group) l_cancel_one [simp]: "x \<in> carrier G \<Longrightarrow> a \<in> carrier G \<Longrightarrow> x \<otimes> a = x \<longleftrightarrow> a = one G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1452 |
by (metis Units_eq Units_l_cancel monoid.r_one monoid_axioms one_closed) |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1453 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1454 |
lemma (in group) r_cancel_one [simp]: "x \<in> carrier G \<Longrightarrow> a \<in> carrier G \<Longrightarrow> a \<otimes> x = x \<longleftrightarrow> a = one G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1455 |
by (metis monoid.l_one monoid_axioms one_closed right_cancel) |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1456 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1457 |
lemma (in group) l_cancel_one' [simp]: "x \<in> carrier G \<Longrightarrow> a \<in> carrier G \<Longrightarrow> x = x \<otimes> a \<longleftrightarrow> a = one G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1458 |
using l_cancel_one by fastforce |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1459 |
|
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1460 |
lemma (in group) r_cancel_one' [simp]: "x \<in> carrier G \<Longrightarrow> a \<in> carrier G \<Longrightarrow> x = a \<otimes> x \<longleftrightarrow> a = one G" |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1461 |
using r_cancel_one by fastforce |
c183a6a69f2d
reorganisation of Algebra: new material from Baillon and Vilhena, removal of duplicate names, elimination of "More_" theories
paulson <lp15@cam.ac.uk>
parents:
68443
diff
changeset
|
1462 |
|
13813 | 1463 |
end |