author | ballarin |
Wed, 30 Apr 2003 10:01:35 +0200 | |
changeset 13936 | d3671b878828 |
parent 13854 | 91c9ab25fece |
child 13940 | c67798653056 |
permissions | -rw-r--r-- |
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(* |
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Title: HOL/Algebra/Group.thy |
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Id: $Id$ |
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Author: Clemens Ballarin, started 4 February 2003 |
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Based on work by Florian Kammueller, L C Paulson and Markus Wenzel. |
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*) |
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header {* Algebraic Structures up to Commutative Groups *} |
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Change to meta simplifier: congruence rules may now have frees as head of term.
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theory Group = FuncSet: |
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axclass number < type |
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instance nat :: number .. |
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instance int :: number .. |
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section {* From Magmas to Groups *} |
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text {* |
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Definitions follow Jacobson, Basic Algebra I, Freeman, 1985; with |
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the exception of \emph{magma} which, following Bourbaki, is a set |
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together with a binary, closed operation. |
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*} |
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subsection {* Definitions *} |
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record 'a semigroup = |
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carrier :: "'a set" |
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mult :: "['a, 'a] => 'a" (infixl "\<otimes>\<index>" 70) |
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record 'a monoid = "'a semigroup" + |
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one :: 'a ("\<one>\<index>") |
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constdefs |
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m_inv :: "[('a, 'm) monoid_scheme, 'a] => 'a" ("inv\<index> _" [81] 80) |
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"m_inv G x == (THE y. y \<in> carrier G & |
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mult G x y = one G & mult G y x = one G)" |
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Units :: "('a, 'm) monoid_scheme => 'a set" |
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"Units G == {y. y \<in> carrier G & |
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(EX x : carrier G. mult G x y = one G & mult G y x = one G)}" |
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consts |
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pow :: "[('a, 'm) monoid_scheme, 'a, 'b::number] => 'a" (infixr "'(^')\<index>" 75) |
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defs (overloaded) |
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nat_pow_def: "pow G a n == nat_rec (one G) (%u b. mult G b a) n" |
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int_pow_def: "pow G a z == |
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let p = nat_rec (one G) (%u b. mult G b a) |
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in if neg z then m_inv G (p (nat (-z))) else p (nat z)" |
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locale magma = struct G + |
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assumes m_closed [intro, simp]: |
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"[| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G" |
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locale semigroup = magma + |
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assumes m_assoc: |
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"[| 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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locale monoid = semigroup + |
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assumes one_closed [intro, simp]: "\<one> \<in> carrier G" |
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and l_one [simp]: "x \<in> carrier G ==> \<one> \<otimes> x = x" |
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and r_one [simp]: "x \<in> carrier G ==> x \<otimes> \<one> = x" |
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lemma monoidI: |
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assumes m_closed: |
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"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G" |
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and one_closed: "one G \<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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mult G (mult G x y) z = mult G x (mult G y z)" |
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and l_one: "!!x. x \<in> carrier G ==> mult G (one G) x = x" |
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and r_one: "!!x. x \<in> carrier G ==> mult G x (one G) = x" |
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shows "monoid G" |
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by (fast intro!: monoid.intro magma.intro semigroup_axioms.intro |
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semigroup.intro monoid_axioms.intro |
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intro: prems) |
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lemma (in monoid) Units_closed [dest]: |
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"x \<in> Units G ==> x \<in> carrier G" |
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by (unfold Units_def) fast |
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lemma (in monoid) inv_unique: |
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assumes eq: "y \<otimes> x = \<one>" "x \<otimes> y' = \<one>" |
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and G: "x \<in> carrier G" "y \<in> carrier G" "y' \<in> carrier G" |
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shows "y = y'" |
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proof - |
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from G eq have "y = y \<otimes> (x \<otimes> y')" by simp |
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also from G have "... = (y \<otimes> x) \<otimes> y'" by (simp add: m_assoc) |
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also from G eq have "... = y'" by simp |
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finally show ?thesis . |
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qed |
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lemma (in monoid) Units_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) |
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apply auto |
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apply (rule theI2, fast) |
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apply (fast intro: inv_unique) |
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apply fast |
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done |
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lemma (in monoid) Units_l_inv: |
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"x \<in> Units G ==> inv x \<otimes> x = \<one>" |
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apply (unfold Units_def m_inv_def) |
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apply auto |
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apply (rule theI2, fast) |
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apply (fast intro: inv_unique) |
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apply fast |
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done |
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lemma (in monoid) Units_r_inv: |
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"x \<in> Units G ==> x \<otimes> inv x = \<one>" |
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apply (unfold Units_def m_inv_def) |
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apply auto |
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apply (rule theI2, fast) |
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apply (fast intro: inv_unique) |
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apply fast |
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done |
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lemma (in monoid) Units_inv_Units [intro, simp]: |
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"x \<in> Units G ==> inv x \<in> Units G" |
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proof - |
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assume x: "x \<in> Units G" |
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show "inv x \<in> Units G" |
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by (auto simp add: Units_def |
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intro: Units_l_inv Units_r_inv x Units_closed [OF x]) |
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qed |
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lemma (in monoid) Units_l_cancel [simp]: |
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"[| x \<in> Units G; y \<in> carrier G; z \<in> carrier G |] ==> |
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(x \<otimes> y = x \<otimes> z) = (y = z)" |
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proof |
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assume eq: "x \<otimes> y = x \<otimes> z" |
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and G: "x \<in> Units G" "y \<in> carrier G" "z \<in> carrier G" |
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then have "(inv x \<otimes> x) \<otimes> y = (inv x \<otimes> x) \<otimes> z" |
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by (simp add: m_assoc Units_closed) |
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with G show "y = z" by (simp add: Units_l_inv) |
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next |
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assume eq: "y = z" |
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and G: "x \<in> Units G" "y \<in> carrier G" "z \<in> carrier G" |
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then show "x \<otimes> y = x \<otimes> z" by simp |
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qed |
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lemma (in monoid) Units_inv_inv [simp]: |
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"x \<in> Units G ==> inv (inv x) = x" |
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proof - |
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assume x: "x \<in> Units G" |
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then have "inv x \<otimes> inv (inv x) = inv x \<otimes> x" |
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by (simp add: Units_l_inv Units_r_inv) |
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with x show ?thesis by (simp add: Units_closed) |
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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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text {* Power *} |
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lemma (in monoid) nat_pow_closed [intro, simp]: |
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"x \<in> carrier G ==> x (^) (n::nat) \<in> carrier G" |
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by (induct n) (simp_all add: nat_pow_def) |
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lemma (in monoid) nat_pow_0 [simp]: |
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"x (^) (0::nat) = \<one>" |
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by (simp add: nat_pow_def) |
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lemma (in monoid) nat_pow_Suc [simp]: |
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"x (^) (Suc n) = x (^) n \<otimes> x" |
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by (simp add: nat_pow_def) |
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lemma (in monoid) nat_pow_one [simp]: |
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"\<one> (^) (n::nat) = \<one>" |
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by (induct n) simp_all |
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lemma (in monoid) nat_pow_mult: |
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"x \<in> carrier G ==> x (^) (n::nat) \<otimes> x (^) m = x (^) (n + m)" |
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by (induct m) (simp_all add: m_assoc [THEN sym]) |
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lemma (in monoid) nat_pow_pow: |
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"x \<in> carrier G ==> (x (^) n) (^) m = x (^) (n * m::nat)" |
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by (induct m) (simp, simp add: nat_pow_mult add_commute) |
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text {* |
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A group is a monoid all of whose elements are invertible. |
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*} |
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locale group = monoid + |
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assumes Units: "carrier G <= Units G" |
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theorem groupI: |
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assumes m_closed [simp]: |
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"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G" |
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and one_closed [simp]: "one G \<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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mult G (mult G x y) z = mult G x (mult G y z)" |
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and l_one [simp]: "!!x. x \<in> carrier G ==> mult G (one G) x = x" |
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and l_inv_ex: "!!x. x \<in> carrier G ==> EX y : carrier G. mult G y x = one G" |
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shows "group G" |
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proof - |
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have l_cancel [simp]: |
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"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
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(mult G x y = mult G x z) = (y = z)" |
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proof |
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fix x y z |
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assume eq: "mult G x y = mult G x z" |
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and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
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with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G" |
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and l_inv: "mult G x_inv x = one G" by fast |
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from G eq xG have "mult G (mult G x_inv x) y = mult G (mult G x_inv x) z" |
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by (simp add: m_assoc) |
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with G show "y = z" by (simp add: l_inv) |
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next |
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fix x y z |
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assume eq: "y = z" |
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and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
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then show "mult G x y = mult G x z" by simp |
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qed |
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have r_one: |
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"!!x. x \<in> carrier G ==> mult G x (one G) = x" |
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proof - |
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fix x |
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assume x: "x \<in> carrier G" |
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with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G" |
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and l_inv: "mult G x_inv x = one G" by fast |
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from x xG have "mult G x_inv (mult G x (one G)) = mult G x_inv x" |
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by (simp add: m_assoc [symmetric] l_inv) |
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with x xG show "mult G x (one G) = x" by simp |
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qed |
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have inv_ex: |
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"!!x. x \<in> carrier G ==> EX y : carrier G. mult G y x = one G & |
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mult G x y = one G" |
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proof - |
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fix x |
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assume x: "x \<in> carrier G" |
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with l_inv_ex obtain y where y: "y \<in> carrier G" |
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and l_inv: "mult G y x = one G" by fast |
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from x y have "mult G y (mult G x y) = mult G y (one G)" |
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by (simp add: m_assoc [symmetric] l_inv r_one) |
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with x y have r_inv: "mult G x y = one G" |
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by simp |
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from x y show "EX y : carrier G. mult G y x = one G & |
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mult G x y = one G" |
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by (fast intro: l_inv r_inv) |
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qed |
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then have carrier_subset_Units: "carrier G <= Units G" |
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by (unfold Units_def) fast |
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show ?thesis |
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by (fast intro!: group.intro magma.intro semigroup_axioms.intro |
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semigroup.intro monoid_axioms.intro group_axioms.intro |
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carrier_subset_Units intro: prems r_one) |
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qed |
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lemma (in monoid) monoid_groupI: |
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assumes l_inv_ex: |
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"!!x. x \<in> carrier G ==> EX y : carrier G. mult G y x = one G" |
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shows "group G" |
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by (rule groupI) (auto intro: m_assoc l_inv_ex) |
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lemma (in group) Units_eq [simp]: |
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"Units G = carrier G" |
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proof |
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show "Units G <= carrier G" by fast |
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next |
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show "carrier G <= Units G" by (rule Units) |
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qed |
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lemma (in group) inv_closed [intro, simp]: |
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"x \<in> carrier G ==> inv x \<in> carrier G" |
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using Units_inv_closed by simp |
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lemma (in group) l_inv: |
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"x \<in> carrier G ==> inv x \<otimes> x = \<one>" |
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using Units_l_inv by simp |
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subsection {* Cancellation Laws and Basic Properties *} |
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lemma (in group) l_cancel [simp]: |
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"[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
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(x \<otimes> y = x \<otimes> z) = (y = z)" |
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using Units_l_inv by simp |
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(* |
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lemma (in group) r_one [simp]: |
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"x \<in> carrier G ==> x \<otimes> \<one> = x" |
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proof - |
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assume x: "x \<in> carrier G" |
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then have "inv x \<otimes> (x \<otimes> \<one>) = inv x \<otimes> x" |
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by (simp add: m_assoc [symmetric] l_inv) |
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with x show ?thesis by simp |
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qed |
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*) |
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lemma (in group) r_inv: |
301 |
"x \<in> carrier G ==> x \<otimes> inv x = \<one>" |
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proof - |
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assume x: "x \<in> carrier G" |
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then have "inv x \<otimes> (x \<otimes> inv x) = inv x \<otimes> \<one>" |
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by (simp add: m_assoc [symmetric] l_inv) |
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with x show ?thesis by (simp del: r_one) |
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qed |
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309 |
lemma (in group) r_cancel [simp]: |
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"[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
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(y \<otimes> x = z \<otimes> x) = (y = z)" |
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proof |
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assume eq: "y \<otimes> x = z \<otimes> x" |
|
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and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
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then have "y \<otimes> (x \<otimes> inv x) = z \<otimes> (x \<otimes> inv x)" |
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by (simp add: m_assoc [symmetric]) |
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317 |
with G show "y = z" by (simp add: r_inv) |
|
318 |
next |
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319 |
assume eq: "y = z" |
|
320 |
and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
|
321 |
then show "y \<otimes> x = z \<otimes> x" by simp |
|
322 |
qed |
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323 |
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13854
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First distributed version of Group and Ring theory.
ballarin
parents:
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|
324 |
lemma (in group) inv_one [simp]: |
91c9ab25fece
First distributed version of Group and Ring theory.
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parents:
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|
325 |
"inv \<one> = \<one>" |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
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diff
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|
326 |
proof - |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
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|
327 |
have "inv \<one> = \<one> \<otimes> (inv \<one>)" by simp |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
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|
328 |
moreover have "... = \<one>" by (simp add: r_inv) |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
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|
329 |
finally show ?thesis . |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
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|
330 |
qed |
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
331 |
|
13813 | 332 |
lemma (in group) inv_inv [simp]: |
333 |
"x \<in> carrier G ==> inv (inv x) = x" |
|
13936 | 334 |
using Units_inv_inv by simp |
335 |
||
336 |
lemma (in group) inv_inj: |
|
337 |
"inj_on (m_inv G) (carrier G)" |
|
338 |
using inv_inj_on_Units by simp |
|
13813 | 339 |
|
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
340 |
lemma (in group) inv_mult_group: |
13813 | 341 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x" |
342 |
proof - |
|
343 |
assume G: "x \<in> carrier G" "y \<in> carrier G" |
|
344 |
then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)" |
|
345 |
by (simp add: m_assoc l_inv) (simp add: m_assoc [symmetric] l_inv) |
|
346 |
with G show ?thesis by simp |
|
347 |
qed |
|
348 |
||
13936 | 349 |
text {* Power *} |
350 |
||
351 |
lemma (in group) int_pow_def2: |
|
352 |
"a (^) (z::int) = (if neg z then inv (a (^) (nat (-z))) else a (^) (nat z))" |
|
353 |
by (simp add: int_pow_def nat_pow_def Let_def) |
|
354 |
||
355 |
lemma (in group) int_pow_0 [simp]: |
|
356 |
"x (^) (0::int) = \<one>" |
|
357 |
by (simp add: int_pow_def2) |
|
358 |
||
359 |
lemma (in group) int_pow_one [simp]: |
|
360 |
"\<one> (^) (z::int) = \<one>" |
|
361 |
by (simp add: int_pow_def2) |
|
362 |
||
13813 | 363 |
subsection {* Substructures *} |
364 |
||
365 |
locale submagma = var H + struct G + |
|
366 |
assumes subset [intro, simp]: "H \<subseteq> carrier G" |
|
367 |
and m_closed [intro, simp]: "[| x \<in> H; y \<in> H |] ==> x \<otimes> y \<in> H" |
|
368 |
||
369 |
declare (in submagma) magma.intro [intro] semigroup.intro [intro] |
|
13936 | 370 |
semigroup_axioms.intro [intro] |
13813 | 371 |
(* |
372 |
alternative definition of submagma |
|
373 |
||
374 |
locale submagma = var H + struct G + |
|
375 |
assumes subset [intro, simp]: "carrier H \<subseteq> carrier G" |
|
376 |
and m_equal [simp]: "mult H = mult G" |
|
377 |
and m_closed [intro, simp]: |
|
378 |
"[| x \<in> carrier H; y \<in> carrier H |] ==> x \<otimes> y \<in> carrier H" |
|
379 |
*) |
|
380 |
||
381 |
lemma submagma_imp_subset: |
|
382 |
"submagma H G ==> H \<subseteq> carrier G" |
|
383 |
by (rule submagma.subset) |
|
384 |
||
385 |
lemma (in submagma) subsetD [dest, simp]: |
|
386 |
"x \<in> H ==> x \<in> carrier G" |
|
387 |
using subset by blast |
|
388 |
||
389 |
lemma (in submagma) magmaI [intro]: |
|
390 |
includes magma G |
|
391 |
shows "magma (G(| carrier := H |))" |
|
392 |
by rule simp |
|
393 |
||
394 |
lemma (in submagma) semigroup_axiomsI [intro]: |
|
395 |
includes semigroup G |
|
396 |
shows "semigroup_axioms (G(| carrier := H |))" |
|
397 |
by rule (simp add: m_assoc) |
|
398 |
||
399 |
lemma (in submagma) semigroupI [intro]: |
|
400 |
includes semigroup G |
|
401 |
shows "semigroup (G(| carrier := H |))" |
|
402 |
using prems by fast |
|
403 |
||
404 |
locale subgroup = submagma H G + |
|
405 |
assumes one_closed [intro, simp]: "\<one> \<in> H" |
|
406 |
and m_inv_closed [intro, simp]: "x \<in> H ==> inv x \<in> H" |
|
407 |
||
408 |
declare (in subgroup) group.intro [intro] |
|
13936 | 409 |
(* |
13817 | 410 |
lemma (in subgroup) l_oneI [intro]: |
411 |
includes l_one G |
|
412 |
shows "l_one (G(| carrier := H |))" |
|
413 |
by rule simp_all |
|
13936 | 414 |
*) |
13813 | 415 |
lemma (in subgroup) group_axiomsI [intro]: |
416 |
includes group G |
|
417 |
shows "group_axioms (G(| carrier := H |))" |
|
13936 | 418 |
by rule (auto intro: l_inv r_inv simp add: Units_def) |
13813 | 419 |
|
420 |
lemma (in subgroup) groupI [intro]: |
|
421 |
includes group G |
|
422 |
shows "group (G(| carrier := H |))" |
|
13936 | 423 |
by (rule groupI) (auto intro: m_assoc l_inv) |
13813 | 424 |
|
425 |
text {* |
|
426 |
Since @{term H} is nonempty, it contains some element @{term x}. Since |
|
427 |
it is closed under inverse, it contains @{text "inv x"}. Since |
|
428 |
it is closed under product, it contains @{text "x \<otimes> inv x = \<one>"}. |
|
429 |
*} |
|
430 |
||
431 |
lemma (in group) one_in_subset: |
|
432 |
"[| 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 |] |
|
433 |
==> \<one> \<in> H" |
|
434 |
by (force simp add: l_inv) |
|
435 |
||
436 |
text {* A characterization of subgroups: closed, non-empty subset. *} |
|
437 |
||
438 |
lemma (in group) subgroupI: |
|
439 |
assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}" |
|
440 |
and inv: "!!a. a \<in> H ==> inv a \<in> H" |
|
441 |
and mult: "!!a b. [|a \<in> H; b \<in> H|] ==> a \<otimes> b \<in> H" |
|
442 |
shows "subgroup H G" |
|
443 |
proof |
|
444 |
from subset and mult show "submagma H G" .. |
|
445 |
next |
|
446 |
have "\<one> \<in> H" by (rule one_in_subset) (auto simp only: prems) |
|
447 |
with inv show "subgroup_axioms H G" |
|
448 |
by (intro subgroup_axioms.intro) simp_all |
|
449 |
qed |
|
450 |
||
451 |
text {* |
|
452 |
Repeat facts of submagmas for subgroups. Necessary??? |
|
453 |
*} |
|
454 |
||
455 |
lemma (in subgroup) subset: |
|
456 |
"H \<subseteq> carrier G" |
|
457 |
.. |
|
458 |
||
459 |
lemma (in subgroup) m_closed: |
|
460 |
"[| x \<in> H; y \<in> H |] ==> x \<otimes> y \<in> H" |
|
461 |
.. |
|
462 |
||
463 |
declare magma.m_closed [simp] |
|
464 |
||
13936 | 465 |
declare monoid.one_closed [iff] group.inv_closed [simp] |
466 |
monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp] |
|
13813 | 467 |
|
468 |
lemma subgroup_nonempty: |
|
469 |
"~ subgroup {} G" |
|
470 |
by (blast dest: subgroup.one_closed) |
|
471 |
||
472 |
lemma (in subgroup) finite_imp_card_positive: |
|
473 |
"finite (carrier G) ==> 0 < card H" |
|
474 |
proof (rule classical) |
|
475 |
have sub: "subgroup H G" using prems .. |
|
476 |
assume fin: "finite (carrier G)" |
|
477 |
and zero: "~ 0 < card H" |
|
478 |
then have "finite H" by (blast intro: finite_subset dest: subset) |
|
479 |
with zero sub have "subgroup {} G" by simp |
|
480 |
with subgroup_nonempty show ?thesis by contradiction |
|
481 |
qed |
|
482 |
||
13936 | 483 |
(* |
484 |
lemma (in monoid) Units_subgroup: |
|
485 |
"subgroup (Units G) G" |
|
486 |
*) |
|
487 |
||
13813 | 488 |
subsection {* Direct Products *} |
489 |
||
490 |
constdefs |
|
13817 | 491 |
DirProdSemigroup :: |
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
492 |
"[('a, 'm) semigroup_scheme, ('b, 'n) semigroup_scheme] |
13817 | 493 |
=> ('a \<times> 'b) semigroup" |
494 |
(infixr "\<times>\<^sub>s" 80) |
|
495 |
"G \<times>\<^sub>s H == (| carrier = carrier G \<times> carrier H, |
|
496 |
mult = (%(xg, xh) (yg, yh). (mult G xg yg, mult H xh yh)) |)" |
|
497 |
||
13936 | 498 |
DirProdGroup :: |
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
499 |
"[('a, 'm) monoid_scheme, ('b, 'n) monoid_scheme] => ('a \<times> 'b) monoid" |
13936 | 500 |
(infixr "\<times>\<^sub>g" 80) |
501 |
"G \<times>\<^sub>g H == (| carrier = carrier (G \<times>\<^sub>s H), |
|
13817 | 502 |
mult = mult (G \<times>\<^sub>s H), |
503 |
one = (one G, one H) |)" |
|
13936 | 504 |
(* |
13813 | 505 |
DirProdGroup :: |
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
506 |
"[('a, 'm) group_scheme, ('b, 'n) group_scheme] => ('a \<times> 'b) group" |
13813 | 507 |
(infixr "\<times>\<^sub>g" 80) |
508 |
"G \<times>\<^sub>g H == (| carrier = carrier (G \<times>\<^sub>m H), |
|
509 |
mult = mult (G \<times>\<^sub>m H), |
|
13817 | 510 |
one = one (G \<times>\<^sub>m H), |
13813 | 511 |
m_inv = (%(g, h). (m_inv G g, m_inv H h)) |)" |
13936 | 512 |
*) |
13813 | 513 |
|
13817 | 514 |
lemma DirProdSemigroup_magma: |
13813 | 515 |
includes magma G + magma H |
13817 | 516 |
shows "magma (G \<times>\<^sub>s H)" |
517 |
by rule (auto simp add: DirProdSemigroup_def) |
|
13813 | 518 |
|
13817 | 519 |
lemma DirProdSemigroup_semigroup_axioms: |
13813 | 520 |
includes semigroup G + semigroup H |
13817 | 521 |
shows "semigroup_axioms (G \<times>\<^sub>s H)" |
522 |
by rule (auto simp add: DirProdSemigroup_def G.m_assoc H.m_assoc) |
|
13813 | 523 |
|
13817 | 524 |
lemma DirProdSemigroup_semigroup: |
13813 | 525 |
includes semigroup G + semigroup H |
13817 | 526 |
shows "semigroup (G \<times>\<^sub>s H)" |
13813 | 527 |
using prems |
528 |
by (fast intro: semigroup.intro |
|
13817 | 529 |
DirProdSemigroup_magma DirProdSemigroup_semigroup_axioms) |
13813 | 530 |
|
531 |
lemma DirProdGroup_magma: |
|
532 |
includes magma G + magma H |
|
533 |
shows "magma (G \<times>\<^sub>g H)" |
|
13817 | 534 |
by rule |
13936 | 535 |
(auto simp add: DirProdGroup_def DirProdSemigroup_def) |
13813 | 536 |
|
537 |
lemma DirProdGroup_semigroup_axioms: |
|
538 |
includes semigroup G + semigroup H |
|
539 |
shows "semigroup_axioms (G \<times>\<^sub>g H)" |
|
540 |
by rule |
|
13936 | 541 |
(auto simp add: DirProdGroup_def DirProdSemigroup_def |
13817 | 542 |
G.m_assoc H.m_assoc) |
13813 | 543 |
|
544 |
lemma DirProdGroup_semigroup: |
|
545 |
includes semigroup G + semigroup H |
|
546 |
shows "semigroup (G \<times>\<^sub>g H)" |
|
547 |
using prems |
|
548 |
by (fast intro: semigroup.intro |
|
549 |
DirProdGroup_magma DirProdGroup_semigroup_axioms) |
|
550 |
||
551 |
(* ... and further lemmas for group ... *) |
|
552 |
||
13817 | 553 |
lemma DirProdGroup_group: |
13813 | 554 |
includes group G + group H |
555 |
shows "group (G \<times>\<^sub>g H)" |
|
13936 | 556 |
by (rule groupI) |
557 |
(auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv |
|
558 |
simp add: DirProdGroup_def DirProdSemigroup_def) |
|
13813 | 559 |
|
560 |
subsection {* Homomorphisms *} |
|
561 |
||
562 |
constdefs |
|
13817 | 563 |
hom :: "[('a, 'c) semigroup_scheme, ('b, 'd) semigroup_scheme] |
564 |
=> ('a => 'b)set" |
|
13813 | 565 |
"hom G H == |
566 |
{h. h \<in> carrier G -> carrier H & |
|
567 |
(\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (mult G x y) = mult H (h x) (h y))}" |
|
568 |
||
569 |
lemma (in semigroup) hom: |
|
570 |
includes semigroup G |
|
571 |
shows "semigroup (| carrier = hom G G, mult = op o |)" |
|
572 |
proof |
|
573 |
show "magma (| carrier = hom G G, mult = op o |)" |
|
574 |
by rule (simp add: Pi_def hom_def) |
|
575 |
next |
|
576 |
show "semigroup_axioms (| carrier = hom G G, mult = op o |)" |
|
577 |
by rule (simp add: o_assoc) |
|
578 |
qed |
|
579 |
||
580 |
lemma hom_mult: |
|
581 |
"[| h \<in> hom G H; x \<in> carrier G; y \<in> carrier G |] |
|
582 |
==> h (mult G x y) = mult H (h x) (h y)" |
|
583 |
by (simp add: hom_def) |
|
584 |
||
585 |
lemma hom_closed: |
|
586 |
"[| h \<in> hom G H; x \<in> carrier G |] ==> h x \<in> carrier H" |
|
587 |
by (auto simp add: hom_def funcset_mem) |
|
588 |
||
589 |
locale group_hom = group G + group H + var h + |
|
590 |
assumes homh: "h \<in> hom G H" |
|
591 |
notes hom_mult [simp] = hom_mult [OF homh] |
|
592 |
and hom_closed [simp] = hom_closed [OF homh] |
|
593 |
||
594 |
lemma (in group_hom) one_closed [simp]: |
|
595 |
"h \<one> \<in> carrier H" |
|
596 |
by simp |
|
597 |
||
598 |
lemma (in group_hom) hom_one [simp]: |
|
599 |
"h \<one> = \<one>\<^sub>2" |
|
600 |
proof - |
|
601 |
have "h \<one> \<otimes>\<^sub>2 \<one>\<^sub>2 = h \<one> \<otimes>\<^sub>2 h \<one>" |
|
602 |
by (simp add: hom_mult [symmetric] del: hom_mult) |
|
603 |
then show ?thesis by (simp del: r_one) |
|
604 |
qed |
|
605 |
||
606 |
lemma (in group_hom) inv_closed [simp]: |
|
607 |
"x \<in> carrier G ==> h (inv x) \<in> carrier H" |
|
608 |
by simp |
|
609 |
||
610 |
lemma (in group_hom) hom_inv [simp]: |
|
611 |
"x \<in> carrier G ==> h (inv x) = inv\<^sub>2 (h x)" |
|
612 |
proof - |
|
613 |
assume x: "x \<in> carrier G" |
|
614 |
then have "h x \<otimes>\<^sub>2 h (inv x) = \<one>\<^sub>2" |
|
615 |
by (simp add: hom_mult [symmetric] G.r_inv del: hom_mult) |
|
616 |
also from x have "... = h x \<otimes>\<^sub>2 inv\<^sub>2 (h x)" |
|
617 |
by (simp add: hom_mult [symmetric] H.r_inv del: hom_mult) |
|
618 |
finally have "h x \<otimes>\<^sub>2 h (inv x) = h x \<otimes>\<^sub>2 inv\<^sub>2 (h x)" . |
|
619 |
with x show ?thesis by simp |
|
620 |
qed |
|
621 |
||
13936 | 622 |
section {* Commutative Structures *} |
623 |
||
624 |
text {* |
|
625 |
Naming convention: multiplicative structures that are commutative |
|
626 |
are called \emph{commutative}, additive structures are called |
|
627 |
\emph{Abelian}. |
|
628 |
*} |
|
13813 | 629 |
|
630 |
subsection {* Definition *} |
|
631 |
||
13936 | 632 |
locale comm_semigroup = semigroup + |
13813 | 633 |
assumes m_comm: "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x" |
634 |
||
13936 | 635 |
lemma (in comm_semigroup) m_lcomm: |
13813 | 636 |
"[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
637 |
x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)" |
|
638 |
proof - |
|
639 |
assume xyz: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" |
|
640 |
from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc) |
|
641 |
also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm) |
|
642 |
also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc) |
|
643 |
finally show ?thesis . |
|
644 |
qed |
|
645 |
||
13936 | 646 |
lemmas (in comm_semigroup) m_ac = m_assoc m_comm m_lcomm |
647 |
||
648 |
locale comm_monoid = comm_semigroup + monoid |
|
13813 | 649 |
|
13936 | 650 |
lemma comm_monoidI: |
651 |
assumes m_closed: |
|
652 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G" |
|
653 |
and one_closed: "one G \<in> carrier G" |
|
654 |
and m_assoc: |
|
655 |
"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
656 |
mult G (mult G x y) z = mult G x (mult G y z)" |
|
657 |
and l_one: "!!x. x \<in> carrier G ==> mult G (one G) x = x" |
|
658 |
and m_comm: |
|
659 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y = mult G y x" |
|
660 |
shows "comm_monoid G" |
|
661 |
using l_one |
|
662 |
by (auto intro!: comm_monoid.intro magma.intro semigroup_axioms.intro |
|
663 |
comm_semigroup_axioms.intro monoid_axioms.intro |
|
664 |
intro: prems simp: m_closed one_closed m_comm) |
|
13817 | 665 |
|
13936 | 666 |
lemma (in monoid) monoid_comm_monoidI: |
667 |
assumes m_comm: |
|
668 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y = mult G y x" |
|
669 |
shows "comm_monoid G" |
|
670 |
by (rule comm_monoidI) (auto intro: m_assoc m_comm) |
|
671 |
(* |
|
672 |
lemma (in comm_monoid) r_one [simp]: |
|
13817 | 673 |
"x \<in> carrier G ==> x \<otimes> \<one> = x" |
674 |
proof - |
|
675 |
assume G: "x \<in> carrier G" |
|
676 |
then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm) |
|
677 |
also from G have "... = x" by simp |
|
678 |
finally show ?thesis . |
|
679 |
qed |
|
13936 | 680 |
*) |
13817 | 681 |
|
13936 | 682 |
lemma (in comm_monoid) nat_pow_distr: |
683 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> |
|
684 |
(x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n" |
|
685 |
by (induct n) (simp, simp add: m_ac) |
|
686 |
||
687 |
locale comm_group = comm_monoid + group |
|
688 |
||
689 |
lemma (in group) group_comm_groupI: |
|
690 |
assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> |
|
691 |
mult G x y = mult G y x" |
|
692 |
shows "comm_group G" |
|
693 |
by (fast intro: comm_group.intro comm_semigroup_axioms.intro |
|
694 |
group.axioms prems) |
|
13817 | 695 |
|
13936 | 696 |
lemma comm_groupI: |
697 |
assumes m_closed: |
|
698 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G" |
|
699 |
and one_closed: "one G \<in> carrier G" |
|
700 |
and m_assoc: |
|
701 |
"!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==> |
|
702 |
mult G (mult G x y) z = mult G x (mult G y z)" |
|
703 |
and m_comm: |
|
704 |
"!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y = mult G y x" |
|
705 |
and l_one: "!!x. x \<in> carrier G ==> mult G (one G) x = x" |
|
706 |
and l_inv_ex: "!!x. x \<in> carrier G ==> EX y : carrier G. mult G y x = one G" |
|
707 |
shows "comm_group G" |
|
708 |
by (fast intro: group.group_comm_groupI groupI prems) |
|
709 |
||
710 |
lemma (in comm_group) inv_mult: |
|
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
711 |
"[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y" |
13936 | 712 |
by (simp add: m_ac inv_mult_group) |
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset
|
713 |
|
13813 | 714 |
end |