src/HOL/Algebra/More_Ring.thy

tuned headers;

(*  Title:      HOL/Algebra/More_Ring.thy
    Author:     Jeremy Avigad
*)

section \<open>More on rings etc.\<close>

theory More_Ring
imports
  Ring
begin

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"
  apply (unfold_locales)
  apply (insert cring_axioms, auto)
  apply (rule trans)
  apply (subgoal_tac "a = (a \<otimes> b) \<otimes> inv b")
  apply assumption
  apply (subst m_assoc)
  apply auto
  apply (unfold Units_def)
  apply auto
  done

lemma (in monoid) inv_char: "x : carrier G \<Longrightarrow> y : carrier G \<Longrightarrow>
    x \<otimes> y = \<one> \<Longrightarrow> y \<otimes> x = \<one> \<Longrightarrow> inv x = y"
  apply (subgoal_tac "x : Units G")
  apply (subgoal_tac "y = inv x \<otimes> \<one>")
  apply simp
  apply (erule subst)
  apply (subst m_assoc [symmetric])
  apply auto
  apply (unfold Units_def)
  apply auto
  done

lemma (in comm_monoid) comm_inv_char: "x : carrier G \<Longrightarrow> y : carrier G \<Longrightarrow>
  x \<otimes> y = \<one> \<Longrightarrow> inv x = y"
  apply (rule inv_char)
  apply auto
  apply (subst m_comm, auto)
  done

lemma (in ring) inv_neg_one [simp]: "inv (\<ominus> \<one>) = \<ominus> \<one>"
  apply (rule inv_char)
  apply (auto simp add: l_minus r_minus)
  done

lemma (in monoid) inv_eq_imp_eq: "x : Units G \<Longrightarrow> y : Units G \<Longrightarrow>
    inv x = inv y \<Longrightarrow> x = y"
  apply (subgoal_tac "inv(inv x) = inv(inv y)")
  apply (subst (asm) Units_inv_inv)+
  apply auto
  done

lemma (in ring) Units_minus_one_closed [intro]: "\<ominus> \<one> : Units R"
  apply (unfold Units_def)
  apply auto
  apply (rule_tac x = "\<ominus> \<one>" in bexI)
  apply auto
  apply (simp add: l_minus r_minus)
  done

lemma (in monoid) inv_one [simp]: "inv \<one> = \<one>"
  apply (rule inv_char)
  apply auto
  done

lemma (in ring) inv_eq_neg_one_eq: "x : Units R \<Longrightarrow> (inv x = \<ominus> \<one>) = (x = \<ominus> \<one>)"
  apply auto
  apply (subst Units_inv_inv [symmetric])
  apply auto
  done

lemma (in monoid) inv_eq_one_eq: "x : Units G \<Longrightarrow> (inv x = \<one>) = (x = \<one>)"
  by (metis Units_inv_inv inv_one)

end