author | haftmann |
Wed, 17 Feb 2016 21:51:56 +0100 | |
changeset 62343 | 24106dc44def |
parent 61952 | 546958347e05 |
permissions | -rw-r--r-- |
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(* Title: HOL/Number_Theory/MiscAlgebra.thy |
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Author: Jeremy Avigad |
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*) |
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section \<open>Things that can be added to the Algebra library\<close> |
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theory MiscAlgebra |
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imports |
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"~~/src/HOL/Algebra/Ring" |
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"~~/src/HOL/Algebra/FiniteProduct" |
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begin |
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subsection \<open>Finiteness stuff\<close> |
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lemma bounded_set1_int [intro]: "finite {(x::int). a < x & x < b & P x}" |
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apply (subgoal_tac "{x. a < x & x < b & P x} <= {a<..<b}") |
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apply (erule finite_subset) |
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apply auto |
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done |
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||
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subsection \<open>The rest is for the algebra libraries\<close> |
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subsubsection \<open>These go in Group.thy\<close> |
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text \<open> |
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Show that the units in any monoid give rise to a group. |
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The file Residues.thy provides some infrastructure to use |
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facts about the unit group within the ring locale. |
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\<close> |
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definition units_of :: "('a, 'b) monoid_scheme => 'a monoid" where |
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"units_of G == (| carrier = Units G, |
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Group.monoid.mult = Group.monoid.mult G, |
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one = one G |)" |
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(* |
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||
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lemma (in monoid) Units_mult_closed [intro]: |
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"x : Units G ==> y : Units G ==> x \<otimes> y : Units G" |
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apply (unfold Units_def) |
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apply (clarsimp) |
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apply (rule_tac x = "xaa \<otimes> xa" in bexI) |
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apply auto |
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apply (subst m_assoc) |
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apply auto |
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apply (subst (2) m_assoc [symmetric]) |
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apply auto |
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apply (subst m_assoc) |
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apply auto |
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apply (subst (2) m_assoc [symmetric]) |
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apply auto |
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done |
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*) |
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lemma (in monoid) units_group: "group(units_of G)" |
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apply (unfold units_of_def) |
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apply (rule groupI) |
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apply auto |
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apply (subst m_assoc) |
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apply auto |
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apply (rule_tac x = "inv x" in bexI) |
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apply auto |
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done |
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lemma (in comm_monoid) units_comm_group: "comm_group(units_of G)" |
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apply (rule group.group_comm_groupI) |
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apply (rule units_group) |
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apply (insert comm_monoid_axioms) |
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apply (unfold units_of_def Units_def comm_monoid_def comm_monoid_axioms_def) |
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apply auto |
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done |
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lemma units_of_carrier: "carrier (units_of G) = Units G" |
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unfolding units_of_def by auto |
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lemma units_of_mult: "mult(units_of G) = mult G" |
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unfolding units_of_def by auto |
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lemma units_of_one: "one(units_of G) = one G" |
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unfolding units_of_def by auto |
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lemma (in monoid) units_of_inv: "x : Units G ==> m_inv (units_of G) x = m_inv G x" |
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apply (rule sym) |
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apply (subst m_inv_def) |
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apply (rule the1_equality) |
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apply (rule ex_ex1I) |
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apply (subst (asm) Units_def) |
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apply auto |
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apply (erule inv_unique) |
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apply auto |
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apply (rule Units_closed) |
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apply (simp_all only: units_of_carrier [symmetric]) |
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apply (insert units_group) |
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apply auto |
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apply (subst units_of_mult [symmetric]) |
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apply (subst units_of_one [symmetric]) |
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apply (erule group.r_inv, assumption) |
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apply (subst units_of_mult [symmetric]) |
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apply (subst units_of_one [symmetric]) |
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apply (erule group.l_inv, assumption) |
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done |
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lemma (in group) inj_on_const_mult: "a: (carrier G) ==> inj_on (%x. a \<otimes> x) (carrier G)" |
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unfolding inj_on_def by auto |
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lemma (in group) surj_const_mult: "a : (carrier G) ==> (%x. a \<otimes> x) ` (carrier G) = (carrier G)" |
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apply (auto simp add: image_def) |
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apply (rule_tac x = "(m_inv G a) \<otimes> x" in bexI) |
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apply auto |
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(* auto should get this. I suppose we need "comm_monoid_simprules" |
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for ac_simps rewriting. *) |
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apply (subst m_assoc [symmetric]) |
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apply auto |
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done |
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lemma (in group) l_cancel_one [simp]: |
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"x : carrier G \<Longrightarrow> a : carrier G \<Longrightarrow> (x \<otimes> a = x) = (a = one G)" |
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apply auto |
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apply (subst l_cancel [symmetric]) |
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prefer 4 |
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apply (erule ssubst) |
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apply auto |
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done |
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lemma (in group) r_cancel_one [simp]: "x : carrier G \<Longrightarrow> a : carrier G \<Longrightarrow> |
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(a \<otimes> x = x) = (a = one G)" |
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apply auto |
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apply (subst r_cancel [symmetric]) |
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prefer 4 |
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apply (erule ssubst) |
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apply auto |
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done |
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(* Is there a better way to do this? *) |
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lemma (in group) l_cancel_one' [simp]: "x : carrier G \<Longrightarrow> a : carrier G \<Longrightarrow> |
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(x = x \<otimes> a) = (a = one G)" |
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apply (subst eq_commute) |
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apply simp |
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done |
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lemma (in group) r_cancel_one' [simp]: "x : carrier G \<Longrightarrow> a : carrier G \<Longrightarrow> |
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(x = a \<otimes> x) = (a = one G)" |
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apply (subst eq_commute) |
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apply simp |
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done |
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(* This should be generalized to arbitrary groups, not just commutative |
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ones, using Lagrange's theorem. *) |
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lemma (in comm_group) power_order_eq_one: |
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assumes fin [simp]: "finite (carrier G)" |
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and a [simp]: "a : carrier G" |
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shows "a (^) card(carrier G) = one G" |
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proof - |
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have "(\<Otimes>x\<in>carrier G. x) = (\<Otimes>x\<in>carrier G. a \<otimes> x)" |
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by (subst (2) finprod_reindex [symmetric], |
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auto simp add: Pi_def inj_on_const_mult surj_const_mult) |
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also have "\<dots> = (\<Otimes>x\<in>carrier G. a) \<otimes> (\<Otimes>x\<in>carrier G. x)" |
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by (auto simp add: finprod_multf Pi_def) |
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also have "(\<Otimes>x\<in>carrier G. a) = a (^) card(carrier G)" |
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by (auto simp add: finprod_const) |
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finally show ?thesis |
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(* uses the preceeding lemma *) |
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by auto |
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qed |
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subsubsection \<open>Miscellaneous\<close> |
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lemma (in cring) field_intro2: "\<zero>\<^bsub>R\<^esub> ~= \<one>\<^bsub>R\<^esub> \<Longrightarrow> \<forall>x \<in> carrier R - {\<zero>\<^bsub>R\<^esub>}. x \<in> Units R \<Longrightarrow> field R" |
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apply (unfold_locales) |
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apply (insert cring_axioms, auto) |
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apply (rule trans) |
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apply (subgoal_tac "a = (a \<otimes> b) \<otimes> inv b") |
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apply assumption |
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apply (subst m_assoc) |
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apply auto |
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apply (unfold Units_def) |
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apply auto |
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done |
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lemma (in monoid) inv_char: "x : carrier G \<Longrightarrow> y : carrier G \<Longrightarrow> |
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x \<otimes> y = \<one> \<Longrightarrow> y \<otimes> x = \<one> \<Longrightarrow> inv x = y" |
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apply (subgoal_tac "x : Units G") |
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apply (subgoal_tac "y = inv x \<otimes> \<one>") |
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apply simp |
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apply (erule subst) |
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apply (subst m_assoc [symmetric]) |
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apply auto |
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apply (unfold Units_def) |
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apply auto |
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done |
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lemma (in comm_monoid) comm_inv_char: "x : carrier G \<Longrightarrow> y : carrier G \<Longrightarrow> |
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x \<otimes> y = \<one> \<Longrightarrow> inv x = y" |
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apply (rule inv_char) |
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apply auto |
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apply (subst m_comm, auto) |
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done |
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lemma (in ring) inv_neg_one [simp]: "inv (\<ominus> \<one>) = \<ominus> \<one>" |
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apply (rule inv_char) |
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apply (auto simp add: l_minus r_minus) |
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done |
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lemma (in monoid) inv_eq_imp_eq: "x : Units G \<Longrightarrow> y : Units G \<Longrightarrow> |
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inv x = inv y \<Longrightarrow> x = y" |
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apply (subgoal_tac "inv(inv x) = inv(inv y)") |
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apply (subst (asm) Units_inv_inv)+ |
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apply auto |
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done |
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lemma (in ring) Units_minus_one_closed [intro]: "\<ominus> \<one> : Units R" |
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apply (unfold Units_def) |
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apply auto |
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apply (rule_tac x = "\<ominus> \<one>" in bexI) |
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apply auto |
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apply (simp add: l_minus r_minus) |
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done |
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lemma (in monoid) inv_one [simp]: "inv \<one> = \<one>" |
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apply (rule inv_char) |
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apply auto |
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done |
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lemma (in ring) inv_eq_neg_one_eq: "x : Units R \<Longrightarrow> (inv x = \<ominus> \<one>) = (x = \<ominus> \<one>)" |
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apply auto |
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apply (subst Units_inv_inv [symmetric]) |
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apply auto |
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done |
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lemma (in monoid) inv_eq_one_eq: "x : Units G \<Longrightarrow> (inv x = \<one>) = (x = \<one>)" |
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by (metis Units_inv_inv inv_one) |
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subsubsection \<open>This goes in FiniteProduct\<close> |
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lemma (in comm_monoid) finprod_UN_disjoint: |
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"finite I \<Longrightarrow> (ALL i:I. finite (A i)) \<longrightarrow> (ALL i:I. ALL j:I. i ~= j \<longrightarrow> |
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(A i) Int (A j) = {}) \<longrightarrow> |
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(ALL i:I. ALL x: (A i). g x : carrier G) \<longrightarrow> |
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finprod G g (UNION I A) = finprod G (%i. finprod G g (A i)) I" |
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apply (induct set: finite) |
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apply force |
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apply clarsimp |
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apply (subst finprod_Un_disjoint) |
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apply blast |
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apply (erule finite_UN_I) |
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apply blast |
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apply (fastforce) |
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apply (auto intro!: funcsetI finprod_closed) |
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done |
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lemma (in comm_monoid) finprod_Union_disjoint: |
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"[| finite C; (ALL A:C. finite A & (ALL x:A. f x : carrier G)); |
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(ALL A:C. ALL B:C. A ~= B --> A Int B = {}) |] |
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==> finprod G f (\<Union>C) = finprod G (finprod G f) C" |
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apply (frule finprod_UN_disjoint [of C id f]) |
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apply auto |
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done |
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lemma (in comm_monoid) finprod_one: |
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"finite A \<Longrightarrow> (\<And>x. x:A \<Longrightarrow> f x = \<one>) \<Longrightarrow> finprod G f A = \<one>" |
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by (induct set: finite) auto |
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(* need better simplification rules for rings *) |
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(* the next one holds more generally for abelian groups *) |
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||
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lemma (in cring) sum_zero_eq_neg: "x : carrier R \<Longrightarrow> y : carrier R \<Longrightarrow> x \<oplus> y = \<zero> \<Longrightarrow> x = \<ominus> y" |
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by (metis minus_equality) |
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lemma (in domain) square_eq_one: |
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fixes x |
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assumes [simp]: "x : carrier R" |
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and "x \<otimes> x = \<one>" |
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shows "x = \<one> | x = \<ominus>\<one>" |
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proof - |
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have "(x \<oplus> \<one>) \<otimes> (x \<oplus> \<ominus> \<one>) = x \<otimes> x \<oplus> \<ominus> \<one>" |
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by (simp add: ring_simprules) |
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also from \<open>x \<otimes> x = \<one>\<close> have "\<dots> = \<zero>" |
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by (simp add: ring_simprules) |
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finally have "(x \<oplus> \<one>) \<otimes> (x \<oplus> \<ominus> \<one>) = \<zero>" . |
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then have "(x \<oplus> \<one>) = \<zero> | (x \<oplus> \<ominus> \<one>) = \<zero>" |
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by (intro integral, auto) |
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then show ?thesis |
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apply auto |
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apply (erule notE) |
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apply (rule sum_zero_eq_neg) |
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apply auto |
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apply (subgoal_tac "x = \<ominus> (\<ominus> \<one>)") |
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apply (simp add: ring_simprules) |
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apply (rule sum_zero_eq_neg) |
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apply auto |
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done |
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qed |
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||
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lemma (in Ring.domain) inv_eq_self: "x : Units R \<Longrightarrow> x = inv x \<Longrightarrow> x = \<one> \<or> x = \<ominus>\<one>" |
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by (metis Units_closed Units_l_inv square_eq_one) |
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text \<open> |
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The following translates theorems about groups to the facts about |
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the units of a ring. (The list should be expanded as more things are |
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needed.) |
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\<close> |
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lemma (in ring) finite_ring_finite_units [intro]: "finite (carrier R) \<Longrightarrow> finite (Units R)" |
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by (rule finite_subset) auto |
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lemma (in monoid) units_of_pow: |
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fixes n :: nat |
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shows "x \<in> Units G \<Longrightarrow> x (^)\<^bsub>units_of G\<^esub> n = x (^)\<^bsub>G\<^esub> n" |
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apply (induct n) |
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apply (auto simp add: units_group group.is_monoid |
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monoid.nat_pow_0 monoid.nat_pow_Suc units_of_one units_of_mult) |
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done |
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lemma (in cring) units_power_order_eq_one: "finite (Units R) \<Longrightarrow> a : Units R |
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\<Longrightarrow> a (^) card(Units R) = \<one>" |
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apply (subst units_of_carrier [symmetric]) |
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apply (subst units_of_one [symmetric]) |
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apply (subst units_of_pow [symmetric]) |
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apply assumption |
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apply (rule comm_group.power_order_eq_one) |
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apply (rule units_comm_group) |
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apply (unfold units_of_def, auto) |
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done |
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end |