src/HOL/Fields.thy
author hoelzl
Fri Mar 22 10:41:43 2013 +0100 (2013-03-22)
changeset 51474 1e9e68247ad1
parent 44921 58eef4843641
child 52435 6646bb548c6b
permissions -rw-r--r--
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
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(*  Title:      HOL/Fields.thy
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    Author:     Gertrud Bauer
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    Author:     Steven Obua
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    Author:     Tobias Nipkow
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    Author:     Lawrence C Paulson
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    Author:     Markus Wenzel
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    Author:     Jeremy Avigad
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*)
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header {* Fields *}
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theory Fields
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imports Rings
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begin
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subsection {* Division rings *}
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text {*
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  A division ring is like a field, but without the commutativity requirement.
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*}
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class inverse =
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  fixes inverse :: "'a \<Rightarrow> 'a"
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    and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "'/" 70)
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class division_ring = ring_1 + inverse +
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  assumes left_inverse [simp]:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
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  assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"
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  assumes divide_inverse: "a / b = a * inverse b"
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begin
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subclass ring_1_no_zero_divisors
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proof
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  fix a b :: 'a
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  assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
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  show "a * b \<noteq> 0"
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  proof
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    assume ab: "a * b = 0"
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    hence "0 = inverse a * (a * b) * inverse b" by simp
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    also have "\<dots> = (inverse a * a) * (b * inverse b)"
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      by (simp only: mult_assoc)
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    also have "\<dots> = 1" using a b by simp
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    finally show False by simp
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  qed
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qed
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lemma nonzero_imp_inverse_nonzero:
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  "a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0"
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proof
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  assume ianz: "inverse a = 0"
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  assume "a \<noteq> 0"
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  hence "1 = a * inverse a" by simp
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  also have "... = 0" by (simp add: ianz)
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  finally have "1 = 0" .
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  thus False by (simp add: eq_commute)
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qed
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lemma inverse_zero_imp_zero:
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  "inverse a = 0 \<Longrightarrow> a = 0"
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apply (rule classical)
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apply (drule nonzero_imp_inverse_nonzero)
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apply auto
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done
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lemma inverse_unique: 
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  assumes ab: "a * b = 1"
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  shows "inverse a = b"
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proof -
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  have "a \<noteq> 0" using ab by (cases "a = 0") simp_all
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  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab)
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  ultimately show ?thesis by (simp add: mult_assoc [symmetric])
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qed
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lemma nonzero_inverse_minus_eq:
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  "a \<noteq> 0 \<Longrightarrow> inverse (- a) = - inverse a"
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by (rule inverse_unique) simp
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lemma nonzero_inverse_inverse_eq:
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  "a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a"
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by (rule inverse_unique) simp
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lemma nonzero_inverse_eq_imp_eq:
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  assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0"
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  shows "a = b"
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proof -
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  from `inverse a = inverse b`
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  have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong)
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  with `a \<noteq> 0` and `b \<noteq> 0` show "a = b"
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    by (simp add: nonzero_inverse_inverse_eq)
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qed
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lemma inverse_1 [simp]: "inverse 1 = 1"
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by (rule inverse_unique) simp
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lemma nonzero_inverse_mult_distrib: 
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  assumes "a \<noteq> 0" and "b \<noteq> 0"
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  shows "inverse (a * b) = inverse b * inverse a"
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proof -
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  have "a * (b * inverse b) * inverse a = 1" using assms by simp
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  hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult_assoc)
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  thus ?thesis by (rule inverse_unique)
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qed
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lemma division_ring_inverse_add:
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  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b"
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by (simp add: algebra_simps)
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lemma division_ring_inverse_diff:
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  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b"
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by (simp add: algebra_simps)
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lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"
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proof
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  assume neq: "b \<noteq> 0"
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  {
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    hence "a = (a / b) * b" by (simp add: divide_inverse mult_assoc)
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    also assume "a / b = 1"
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    finally show "a = b" by simp
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  next
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    assume "a = b"
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    with neq show "a / b = 1" by (simp add: divide_inverse)
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  }
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qed
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lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"
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by (simp add: divide_inverse)
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lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
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by (simp add: divide_inverse)
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lemma divide_zero_left [simp]: "0 / a = 0"
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by (simp add: divide_inverse)
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lemma inverse_eq_divide: "inverse a = 1 / a"
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by (simp add: divide_inverse)
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lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
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by (simp add: divide_inverse algebra_simps)
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lemma divide_1 [simp]: "a / 1 = a"
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  by (simp add: divide_inverse)
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lemma times_divide_eq_right [simp]: "a * (b / c) = (a * b) / c"
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  by (simp add: divide_inverse mult_assoc)
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lemma minus_divide_left: "- (a / b) = (-a) / b"
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  by (simp add: divide_inverse)
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lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a / b) = a / (- b)"
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  by (simp add: divide_inverse nonzero_inverse_minus_eq)
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lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a) / (-b) = a / b"
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  by (simp add: divide_inverse nonzero_inverse_minus_eq)
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lemma divide_minus_left [simp, no_atp]: "(-a) / b = - (a / b)"
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  by (simp add: divide_inverse)
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lemma diff_divide_distrib: "(a - b) / c = a / c - b / c"
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  by (simp add: diff_minus add_divide_distrib)
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lemma nonzero_eq_divide_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b"
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proof -
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  assume [simp]: "c \<noteq> 0"
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  have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp
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  also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult_assoc)
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  finally show ?thesis .
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qed
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lemma nonzero_divide_eq_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> b / c = a \<longleftrightarrow> b = a * c"
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proof -
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  assume [simp]: "c \<noteq> 0"
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  have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp
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  also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult_assoc) 
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  finally show ?thesis .
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qed
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lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a"
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  by (simp add: divide_inverse mult_assoc)
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lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"
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  by (drule sym) (simp add: divide_inverse mult_assoc)
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end
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class division_ring_inverse_zero = division_ring +
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  assumes inverse_zero [simp]: "inverse 0 = 0"
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begin
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lemma divide_zero [simp]:
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  "a / 0 = 0"
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  by (simp add: divide_inverse)
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lemma divide_self_if [simp]:
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  "a / a = (if a = 0 then 0 else 1)"
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  by simp
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lemma inverse_nonzero_iff_nonzero [simp]:
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  "inverse a = 0 \<longleftrightarrow> a = 0"
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  by rule (fact inverse_zero_imp_zero, simp)
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lemma inverse_minus_eq [simp]:
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  "inverse (- a) = - inverse a"
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proof cases
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  assume "a=0" thus ?thesis by simp
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next
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  assume "a\<noteq>0" 
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  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
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qed
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lemma inverse_inverse_eq [simp]:
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  "inverse (inverse a) = a"
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proof cases
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  assume "a=0" thus ?thesis by simp
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next
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  assume "a\<noteq>0" 
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  thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
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qed
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lemma inverse_eq_imp_eq:
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  "inverse a = inverse b \<Longrightarrow> a = b"
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  by (drule arg_cong [where f="inverse"], simp)
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lemma inverse_eq_iff_eq [simp]:
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  "inverse a = inverse b \<longleftrightarrow> a = b"
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  by (force dest!: inverse_eq_imp_eq)
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end
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subsection {* Fields *}
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class field = comm_ring_1 + inverse +
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  assumes field_inverse: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
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  assumes field_divide_inverse: "a / b = a * inverse b"
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begin
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subclass division_ring
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proof
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  fix a :: 'a
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  assume "a \<noteq> 0"
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  thus "inverse a * a = 1" by (rule field_inverse)
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  thus "a * inverse a = 1" by (simp only: mult_commute)
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next
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  fix a b :: 'a
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  show "a / b = a * inverse b" by (rule field_divide_inverse)
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qed
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subclass idom ..
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text{*There is no slick version using division by zero.*}
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lemma inverse_add:
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  "[| a \<noteq> 0;  b \<noteq> 0 |]
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   ==> inverse a + inverse b = (a + b) * inverse a * inverse b"
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by (simp add: division_ring_inverse_add mult_ac)
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lemma nonzero_mult_divide_mult_cancel_left [simp, no_atp]:
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assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/b"
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proof -
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  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
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    by (simp add: divide_inverse nonzero_inverse_mult_distrib)
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  also have "... =  a * inverse b * (inverse c * c)"
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    by (simp only: mult_ac)
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  also have "... =  a * inverse b" by simp
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    finally show ?thesis by (simp add: divide_inverse)
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qed
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lemma nonzero_mult_divide_mult_cancel_right [simp, no_atp]:
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  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (b * c) = a / b"
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by (simp add: mult_commute [of _ c])
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lemma times_divide_eq_left [simp]: "(b / c) * a = (b * a) / c"
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  by (simp add: divide_inverse mult_ac)
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text{*It's not obvious whether @{text times_divide_eq} should be
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  simprules or not. Their effect is to gather terms into one big
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  fraction, like a*b*c / x*y*z. The rationale for that is unclear, but
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  many proofs seem to need them.*}
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lemmas times_divide_eq [no_atp] = times_divide_eq_right times_divide_eq_left
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lemma add_frac_eq:
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  assumes "y \<noteq> 0" and "z \<noteq> 0"
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  shows "x / y + w / z = (x * z + w * y) / (y * z)"
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proof -
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  have "x / y + w / z = (x * z) / (y * z) + (y * w) / (y * z)"
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    using assms by simp
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  also have "\<dots> = (x * z + y * w) / (y * z)"
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    by (simp only: add_divide_distrib)
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  finally show ?thesis
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    by (simp only: mult_commute)
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qed
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text{*Special Cancellation Simprules for Division*}
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lemma nonzero_mult_divide_cancel_right [simp, no_atp]:
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  "b \<noteq> 0 \<Longrightarrow> a * b / b = a"
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  using nonzero_mult_divide_mult_cancel_right [of 1 b a] by simp
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lemma nonzero_mult_divide_cancel_left [simp, no_atp]:
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  "a \<noteq> 0 \<Longrightarrow> a * b / a = b"
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using nonzero_mult_divide_mult_cancel_left [of 1 a b] by simp
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lemma nonzero_divide_mult_cancel_right [simp, no_atp]:
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  "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> b / (a * b) = 1 / a"
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using nonzero_mult_divide_mult_cancel_right [of a b 1] by simp
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lemma nonzero_divide_mult_cancel_left [simp, no_atp]:
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  "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / (a * b) = 1 / b"
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using nonzero_mult_divide_mult_cancel_left [of b a 1] by simp
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lemma nonzero_mult_divide_mult_cancel_left2 [simp, no_atp]:
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  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (c * a) / (b * c) = a / b"
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using nonzero_mult_divide_mult_cancel_left [of b c a] by (simp add: mult_ac)
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lemma nonzero_mult_divide_mult_cancel_right2 [simp, no_atp]:
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  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (c * b) = a / b"
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using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: mult_ac)
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lemma add_divide_eq_iff [field_simps]:
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  "z \<noteq> 0 \<Longrightarrow> x + y / z = (z * x + y) / z"
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   320
  by (simp add: add_divide_distrib)
huffman@30630
   321
haftmann@36348
   322
lemma divide_add_eq_iff [field_simps]:
huffman@30630
   323
  "z \<noteq> 0 \<Longrightarrow> x / z + y = (x + z * y) / z"
haftmann@36301
   324
  by (simp add: add_divide_distrib)
huffman@30630
   325
haftmann@36348
   326
lemma diff_divide_eq_iff [field_simps]:
huffman@30630
   327
  "z \<noteq> 0 \<Longrightarrow> x - y / z = (z * x - y) / z"
haftmann@36301
   328
  by (simp add: diff_divide_distrib)
huffman@30630
   329
haftmann@36348
   330
lemma divide_diff_eq_iff [field_simps]:
huffman@30630
   331
  "z \<noteq> 0 \<Longrightarrow> x / z - y = (x - z * y) / z"
haftmann@36301
   332
  by (simp add: diff_divide_distrib)
huffman@30630
   333
huffman@30630
   334
lemma diff_frac_eq:
huffman@30630
   335
  "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y - w / z = (x * z - w * y) / (y * z)"
haftmann@36348
   336
  by (simp add: field_simps)
huffman@30630
   337
huffman@30630
   338
lemma frac_eq_eq:
huffman@30630
   339
  "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)"
haftmann@36348
   340
  by (simp add: field_simps)
haftmann@36348
   341
haftmann@36348
   342
end
haftmann@36348
   343
haftmann@36348
   344
class field_inverse_zero = field +
haftmann@36348
   345
  assumes field_inverse_zero: "inverse 0 = 0"
haftmann@36348
   346
begin
haftmann@36348
   347
haftmann@36348
   348
subclass division_ring_inverse_zero proof
haftmann@36348
   349
qed (fact field_inverse_zero)
haftmann@25230
   350
paulson@14270
   351
text{*This version builds in division by zero while also re-orienting
paulson@14270
   352
      the right-hand side.*}
paulson@14270
   353
lemma inverse_mult_distrib [simp]:
haftmann@36409
   354
  "inverse (a * b) = inverse a * inverse b"
haftmann@36409
   355
proof cases
haftmann@36409
   356
  assume "a \<noteq> 0 & b \<noteq> 0" 
haftmann@36409
   357
  thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_ac)
haftmann@36409
   358
next
haftmann@36409
   359
  assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
haftmann@36409
   360
  thus ?thesis by force
haftmann@36409
   361
qed
paulson@14270
   362
paulson@14365
   363
lemma inverse_divide [simp]:
haftmann@36409
   364
  "inverse (a / b) = b / a"
haftmann@36301
   365
  by (simp add: divide_inverse mult_commute)
paulson@14365
   366
wenzelm@23389
   367
haftmann@36301
   368
text {* Calculations with fractions *}
avigad@16775
   369
nipkow@23413
   370
text{* There is a whole bunch of simp-rules just for class @{text
nipkow@23413
   371
field} but none for class @{text field} and @{text nonzero_divides}
nipkow@23413
   372
because the latter are covered by a simproc. *}
nipkow@23413
   373
nipkow@23413
   374
lemma mult_divide_mult_cancel_left:
haftmann@36409
   375
  "c \<noteq> 0 \<Longrightarrow> (c * a) / (c * b) = a / b"
haftmann@21328
   376
apply (cases "b = 0")
huffman@35216
   377
apply simp_all
paulson@14277
   378
done
paulson@14277
   379
nipkow@23413
   380
lemma mult_divide_mult_cancel_right:
haftmann@36409
   381
  "c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b"
haftmann@21328
   382
apply (cases "b = 0")
huffman@35216
   383
apply simp_all
paulson@14321
   384
done
nipkow@23413
   385
haftmann@36409
   386
lemma divide_divide_eq_right [simp, no_atp]:
haftmann@36409
   387
  "a / (b / c) = (a * c) / b"
haftmann@36409
   388
  by (simp add: divide_inverse mult_ac)
paulson@14288
   389
haftmann@36409
   390
lemma divide_divide_eq_left [simp, no_atp]:
haftmann@36409
   391
  "(a / b) / c = a / (b * c)"
haftmann@36409
   392
  by (simp add: divide_inverse mult_assoc)
paulson@14288
   393
wenzelm@23389
   394
haftmann@36301
   395
text {*Special Cancellation Simprules for Division*}
paulson@15234
   396
haftmann@36409
   397
lemma mult_divide_mult_cancel_left_if [simp,no_atp]:
haftmann@36409
   398
  shows "(c * a) / (c * b) = (if c = 0 then 0 else a / b)"
haftmann@36409
   399
  by (simp add: mult_divide_mult_cancel_left)
nipkow@23413
   400
paulson@15234
   401
haftmann@36301
   402
text {* Division and Unary Minus *}
paulson@14293
   403
haftmann@36409
   404
lemma minus_divide_right:
haftmann@36409
   405
  "- (a / b) = a / - b"
haftmann@36409
   406
  by (simp add: divide_inverse)
paulson@14430
   407
blanchet@35828
   408
lemma divide_minus_right [simp, no_atp]:
haftmann@36409
   409
  "a / - b = - (a / b)"
haftmann@36409
   410
  by (simp add: divide_inverse)
huffman@30630
   411
huffman@30630
   412
lemma minus_divide_divide:
haftmann@36409
   413
  "(- a) / (- b) = a / b"
haftmann@21328
   414
apply (cases "b=0", simp) 
paulson@14293
   415
apply (simp add: nonzero_minus_divide_divide) 
paulson@14293
   416
done
paulson@14293
   417
nipkow@23482
   418
lemma eq_divide_eq:
haftmann@36409
   419
  "a = b / c \<longleftrightarrow> (if c \<noteq> 0 then a * c = b else a = 0)"
haftmann@36409
   420
  by (simp add: nonzero_eq_divide_eq)
nipkow@23482
   421
nipkow@23482
   422
lemma divide_eq_eq:
haftmann@36409
   423
  "b / c = a \<longleftrightarrow> (if c \<noteq> 0 then b = a * c else a = 0)"
haftmann@36409
   424
  by (force simp add: nonzero_divide_eq_eq)
paulson@14293
   425
haftmann@36301
   426
lemma inverse_eq_1_iff [simp]:
haftmann@36409
   427
  "inverse x = 1 \<longleftrightarrow> x = 1"
haftmann@36409
   428
  by (insert inverse_eq_iff_eq [of x 1], simp) 
wenzelm@23389
   429
haftmann@36409
   430
lemma divide_eq_0_iff [simp, no_atp]:
haftmann@36409
   431
  "a / b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
haftmann@36409
   432
  by (simp add: divide_inverse)
haftmann@36301
   433
haftmann@36409
   434
lemma divide_cancel_right [simp, no_atp]:
haftmann@36409
   435
  "a / c = b / c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@36409
   436
  apply (cases "c=0", simp)
haftmann@36409
   437
  apply (simp add: divide_inverse)
haftmann@36409
   438
  done
haftmann@36301
   439
haftmann@36409
   440
lemma divide_cancel_left [simp, no_atp]:
haftmann@36409
   441
  "c / a = c / b \<longleftrightarrow> c = 0 \<or> a = b" 
haftmann@36409
   442
  apply (cases "c=0", simp)
haftmann@36409
   443
  apply (simp add: divide_inverse)
haftmann@36409
   444
  done
haftmann@36301
   445
haftmann@36409
   446
lemma divide_eq_1_iff [simp, no_atp]:
haftmann@36409
   447
  "a / b = 1 \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
haftmann@36409
   448
  apply (cases "b=0", simp)
haftmann@36409
   449
  apply (simp add: right_inverse_eq)
haftmann@36409
   450
  done
haftmann@36301
   451
haftmann@36409
   452
lemma one_eq_divide_iff [simp, no_atp]:
haftmann@36409
   453
  "1 = a / b \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
haftmann@36409
   454
  by (simp add: eq_commute [of 1])
haftmann@36409
   455
haftmann@36719
   456
lemma times_divide_times_eq:
haftmann@36719
   457
  "(x / y) * (z / w) = (x * z) / (y * w)"
haftmann@36719
   458
  by simp
haftmann@36719
   459
haftmann@36719
   460
lemma add_frac_num:
haftmann@36719
   461
  "y \<noteq> 0 \<Longrightarrow> x / y + z = (x + z * y) / y"
haftmann@36719
   462
  by (simp add: add_divide_distrib)
haftmann@36719
   463
haftmann@36719
   464
lemma add_num_frac:
haftmann@36719
   465
  "y \<noteq> 0 \<Longrightarrow> z + x / y = (x + z * y) / y"
haftmann@36719
   466
  by (simp add: add_divide_distrib add.commute)
haftmann@36719
   467
haftmann@36409
   468
end
haftmann@36301
   469
haftmann@36301
   470
huffman@44064
   471
subsection {* Ordered fields *}
haftmann@36301
   472
haftmann@36301
   473
class linordered_field = field + linordered_idom
haftmann@36301
   474
begin
paulson@14268
   475
paulson@14277
   476
lemma positive_imp_inverse_positive: 
haftmann@36301
   477
  assumes a_gt_0: "0 < a" 
haftmann@36301
   478
  shows "0 < inverse a"
nipkow@23482
   479
proof -
paulson@14268
   480
  have "0 < a * inverse a" 
haftmann@36301
   481
    by (simp add: a_gt_0 [THEN less_imp_not_eq2])
paulson@14268
   482
  thus "0 < inverse a" 
haftmann@36301
   483
    by (simp add: a_gt_0 [THEN less_not_sym] zero_less_mult_iff)
nipkow@23482
   484
qed
paulson@14268
   485
paulson@14277
   486
lemma negative_imp_inverse_negative:
haftmann@36301
   487
  "a < 0 \<Longrightarrow> inverse a < 0"
haftmann@36301
   488
  by (insert positive_imp_inverse_positive [of "-a"], 
haftmann@36301
   489
    simp add: nonzero_inverse_minus_eq less_imp_not_eq)
paulson@14268
   490
paulson@14268
   491
lemma inverse_le_imp_le:
haftmann@36301
   492
  assumes invle: "inverse a \<le> inverse b" and apos: "0 < a"
haftmann@36301
   493
  shows "b \<le> a"
nipkow@23482
   494
proof (rule classical)
paulson@14268
   495
  assume "~ b \<le> a"
nipkow@23482
   496
  hence "a < b"  by (simp add: linorder_not_le)
haftmann@36301
   497
  hence bpos: "0 < b"  by (blast intro: apos less_trans)
paulson@14268
   498
  hence "a * inverse a \<le> a * inverse b"
haftmann@36301
   499
    by (simp add: apos invle less_imp_le mult_left_mono)
paulson@14268
   500
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
haftmann@36301
   501
    by (simp add: bpos less_imp_le mult_right_mono)
haftmann@36301
   502
  thus "b \<le> a"  by (simp add: mult_assoc apos bpos less_imp_not_eq2)
nipkow@23482
   503
qed
paulson@14268
   504
paulson@14277
   505
lemma inverse_positive_imp_positive:
haftmann@36301
   506
  assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
haftmann@36301
   507
  shows "0 < a"
wenzelm@23389
   508
proof -
paulson@14277
   509
  have "0 < inverse (inverse a)"
wenzelm@23389
   510
    using inv_gt_0 by (rule positive_imp_inverse_positive)
paulson@14277
   511
  thus "0 < a"
wenzelm@23389
   512
    using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
   513
qed
paulson@14277
   514
haftmann@36301
   515
lemma inverse_negative_imp_negative:
haftmann@36301
   516
  assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0"
haftmann@36301
   517
  shows "a < 0"
haftmann@36301
   518
proof -
haftmann@36301
   519
  have "inverse (inverse a) < 0"
haftmann@36301
   520
    using inv_less_0 by (rule negative_imp_inverse_negative)
haftmann@36301
   521
  thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
haftmann@36301
   522
qed
haftmann@36301
   523
haftmann@36301
   524
lemma linordered_field_no_lb:
haftmann@36301
   525
  "\<forall>x. \<exists>y. y < x"
haftmann@36301
   526
proof
haftmann@36301
   527
  fix x::'a
haftmann@36301
   528
  have m1: "- (1::'a) < 0" by simp
haftmann@36301
   529
  from add_strict_right_mono[OF m1, where c=x] 
haftmann@36301
   530
  have "(- 1) + x < x" by simp
haftmann@36301
   531
  thus "\<exists>y. y < x" by blast
haftmann@36301
   532
qed
haftmann@36301
   533
haftmann@36301
   534
lemma linordered_field_no_ub:
haftmann@36301
   535
  "\<forall> x. \<exists>y. y > x"
haftmann@36301
   536
proof
haftmann@36301
   537
  fix x::'a
haftmann@36301
   538
  have m1: " (1::'a) > 0" by simp
haftmann@36301
   539
  from add_strict_right_mono[OF m1, where c=x] 
haftmann@36301
   540
  have "1 + x > x" by simp
haftmann@36301
   541
  thus "\<exists>y. y > x" by blast
haftmann@36301
   542
qed
haftmann@36301
   543
haftmann@36301
   544
lemma less_imp_inverse_less:
haftmann@36301
   545
  assumes less: "a < b" and apos:  "0 < a"
haftmann@36301
   546
  shows "inverse b < inverse a"
haftmann@36301
   547
proof (rule ccontr)
haftmann@36301
   548
  assume "~ inverse b < inverse a"
haftmann@36301
   549
  hence "inverse a \<le> inverse b" by simp
haftmann@36301
   550
  hence "~ (a < b)"
haftmann@36301
   551
    by (simp add: not_less inverse_le_imp_le [OF _ apos])
haftmann@36301
   552
  thus False by (rule notE [OF _ less])
haftmann@36301
   553
qed
haftmann@36301
   554
haftmann@36301
   555
lemma inverse_less_imp_less:
haftmann@36301
   556
  "inverse a < inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b < a"
haftmann@36301
   557
apply (simp add: less_le [of "inverse a"] less_le [of "b"])
haftmann@36301
   558
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
haftmann@36301
   559
done
haftmann@36301
   560
haftmann@36301
   561
text{*Both premises are essential. Consider -1 and 1.*}
haftmann@36301
   562
lemma inverse_less_iff_less [simp,no_atp]:
haftmann@36301
   563
  "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
haftmann@36301
   564
  by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
haftmann@36301
   565
haftmann@36301
   566
lemma le_imp_inverse_le:
haftmann@36301
   567
  "a \<le> b \<Longrightarrow> 0 < a \<Longrightarrow> inverse b \<le> inverse a"
haftmann@36301
   568
  by (force simp add: le_less less_imp_inverse_less)
haftmann@36301
   569
haftmann@36301
   570
lemma inverse_le_iff_le [simp,no_atp]:
haftmann@36301
   571
  "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
haftmann@36301
   572
  by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
haftmann@36301
   573
haftmann@36301
   574
haftmann@36301
   575
text{*These results refer to both operands being negative.  The opposite-sign
haftmann@36301
   576
case is trivial, since inverse preserves signs.*}
haftmann@36301
   577
lemma inverse_le_imp_le_neg:
haftmann@36301
   578
  "inverse a \<le> inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a"
haftmann@36301
   579
apply (rule classical) 
haftmann@36301
   580
apply (subgoal_tac "a < 0") 
haftmann@36301
   581
 prefer 2 apply force
haftmann@36301
   582
apply (insert inverse_le_imp_le [of "-b" "-a"])
haftmann@36301
   583
apply (simp add: nonzero_inverse_minus_eq) 
haftmann@36301
   584
done
haftmann@36301
   585
haftmann@36301
   586
lemma less_imp_inverse_less_neg:
haftmann@36301
   587
   "a < b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b < inverse a"
haftmann@36301
   588
apply (subgoal_tac "a < 0") 
haftmann@36301
   589
 prefer 2 apply (blast intro: less_trans) 
haftmann@36301
   590
apply (insert less_imp_inverse_less [of "-b" "-a"])
haftmann@36301
   591
apply (simp add: nonzero_inverse_minus_eq) 
haftmann@36301
   592
done
haftmann@36301
   593
haftmann@36301
   594
lemma inverse_less_imp_less_neg:
haftmann@36301
   595
   "inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a"
haftmann@36301
   596
apply (rule classical) 
haftmann@36301
   597
apply (subgoal_tac "a < 0") 
haftmann@36301
   598
 prefer 2
haftmann@36301
   599
 apply force
haftmann@36301
   600
apply (insert inverse_less_imp_less [of "-b" "-a"])
haftmann@36301
   601
apply (simp add: nonzero_inverse_minus_eq) 
haftmann@36301
   602
done
haftmann@36301
   603
haftmann@36301
   604
lemma inverse_less_iff_less_neg [simp,no_atp]:
haftmann@36301
   605
  "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
haftmann@36301
   606
apply (insert inverse_less_iff_less [of "-b" "-a"])
haftmann@36301
   607
apply (simp del: inverse_less_iff_less 
haftmann@36301
   608
            add: nonzero_inverse_minus_eq)
haftmann@36301
   609
done
haftmann@36301
   610
haftmann@36301
   611
lemma le_imp_inverse_le_neg:
haftmann@36301
   612
  "a \<le> b \<Longrightarrow> b < 0 ==> inverse b \<le> inverse a"
haftmann@36301
   613
  by (force simp add: le_less less_imp_inverse_less_neg)
haftmann@36301
   614
haftmann@36301
   615
lemma inverse_le_iff_le_neg [simp,no_atp]:
haftmann@36301
   616
  "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
haftmann@36301
   617
  by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
haftmann@36301
   618
huffman@36774
   619
lemma one_less_inverse:
huffman@36774
   620
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> 1 < inverse a"
huffman@36774
   621
  using less_imp_inverse_less [of a 1, unfolded inverse_1] .
huffman@36774
   622
huffman@36774
   623
lemma one_le_inverse:
huffman@36774
   624
  "0 < a \<Longrightarrow> a \<le> 1 \<Longrightarrow> 1 \<le> inverse a"
huffman@36774
   625
  using le_imp_inverse_le [of a 1, unfolded inverse_1] .
huffman@36774
   626
haftmann@36348
   627
lemma pos_le_divide_eq [field_simps]: "0 < c ==> (a \<le> b/c) = (a*c \<le> b)"
haftmann@36301
   628
proof -
haftmann@36301
   629
  assume less: "0<c"
haftmann@36301
   630
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
haftmann@36304
   631
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   632
  also have "... = (a*c \<le> b)"
haftmann@36301
   633
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
haftmann@36301
   634
  finally show ?thesis .
haftmann@36301
   635
qed
haftmann@36301
   636
haftmann@36348
   637
lemma neg_le_divide_eq [field_simps]: "c < 0 ==> (a \<le> b/c) = (b \<le> a*c)"
haftmann@36301
   638
proof -
haftmann@36301
   639
  assume less: "c<0"
haftmann@36301
   640
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
haftmann@36304
   641
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   642
  also have "... = (b \<le> a*c)"
haftmann@36301
   643
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
haftmann@36301
   644
  finally show ?thesis .
haftmann@36301
   645
qed
haftmann@36301
   646
haftmann@36348
   647
lemma pos_less_divide_eq [field_simps]:
haftmann@36301
   648
     "0 < c ==> (a < b/c) = (a*c < b)"
haftmann@36301
   649
proof -
haftmann@36301
   650
  assume less: "0<c"
haftmann@36301
   651
  hence "(a < b/c) = (a*c < (b/c)*c)"
haftmann@36304
   652
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   653
  also have "... = (a*c < b)"
haftmann@36301
   654
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
haftmann@36301
   655
  finally show ?thesis .
haftmann@36301
   656
qed
haftmann@36301
   657
haftmann@36348
   658
lemma neg_less_divide_eq [field_simps]:
haftmann@36301
   659
 "c < 0 ==> (a < b/c) = (b < a*c)"
haftmann@36301
   660
proof -
haftmann@36301
   661
  assume less: "c<0"
haftmann@36301
   662
  hence "(a < b/c) = ((b/c)*c < a*c)"
haftmann@36304
   663
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   664
  also have "... = (b < a*c)"
haftmann@36301
   665
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
haftmann@36301
   666
  finally show ?thesis .
haftmann@36301
   667
qed
haftmann@36301
   668
haftmann@36348
   669
lemma pos_divide_less_eq [field_simps]:
haftmann@36301
   670
     "0 < c ==> (b/c < a) = (b < a*c)"
haftmann@36301
   671
proof -
haftmann@36301
   672
  assume less: "0<c"
haftmann@36301
   673
  hence "(b/c < a) = ((b/c)*c < a*c)"
haftmann@36304
   674
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   675
  also have "... = (b < a*c)"
haftmann@36301
   676
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
haftmann@36301
   677
  finally show ?thesis .
haftmann@36301
   678
qed
haftmann@36301
   679
haftmann@36348
   680
lemma neg_divide_less_eq [field_simps]:
haftmann@36301
   681
 "c < 0 ==> (b/c < a) = (a*c < b)"
haftmann@36301
   682
proof -
haftmann@36301
   683
  assume less: "c<0"
haftmann@36301
   684
  hence "(b/c < a) = (a*c < (b/c)*c)"
haftmann@36304
   685
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   686
  also have "... = (a*c < b)"
haftmann@36301
   687
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
haftmann@36301
   688
  finally show ?thesis .
haftmann@36301
   689
qed
haftmann@36301
   690
haftmann@36348
   691
lemma pos_divide_le_eq [field_simps]: "0 < c ==> (b/c \<le> a) = (b \<le> a*c)"
haftmann@36301
   692
proof -
haftmann@36301
   693
  assume less: "0<c"
haftmann@36301
   694
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
haftmann@36304
   695
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   696
  also have "... = (b \<le> a*c)"
haftmann@36301
   697
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
haftmann@36301
   698
  finally show ?thesis .
haftmann@36301
   699
qed
haftmann@36301
   700
haftmann@36348
   701
lemma neg_divide_le_eq [field_simps]: "c < 0 ==> (b/c \<le> a) = (a*c \<le> b)"
haftmann@36301
   702
proof -
haftmann@36301
   703
  assume less: "c<0"
haftmann@36301
   704
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
haftmann@36304
   705
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   706
  also have "... = (a*c \<le> b)"
haftmann@36301
   707
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
haftmann@36301
   708
  finally show ?thesis .
haftmann@36301
   709
qed
haftmann@36301
   710
haftmann@36301
   711
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
haftmann@36301
   712
of positivity/negativity needed for @{text field_simps}. Have not added @{text
haftmann@36301
   713
sign_simps} to @{text field_simps} because the former can lead to case
haftmann@36301
   714
explosions. *}
haftmann@36301
   715
haftmann@36348
   716
lemmas sign_simps [no_atp] = algebra_simps
haftmann@36348
   717
  zero_less_mult_iff mult_less_0_iff
haftmann@36348
   718
haftmann@36348
   719
lemmas (in -) sign_simps [no_atp] = algebra_simps
haftmann@36301
   720
  zero_less_mult_iff mult_less_0_iff
haftmann@36301
   721
haftmann@36301
   722
(* Only works once linear arithmetic is installed:
haftmann@36301
   723
text{*An example:*}
haftmann@36301
   724
lemma fixes a b c d e f :: "'a::linordered_field"
haftmann@36301
   725
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
haftmann@36301
   726
 ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
haftmann@36301
   727
 ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
haftmann@36301
   728
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
haftmann@36301
   729
 prefer 2 apply(simp add:sign_simps)
haftmann@36301
   730
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
haftmann@36301
   731
 prefer 2 apply(simp add:sign_simps)
haftmann@36301
   732
apply(simp add:field_simps)
haftmann@36301
   733
done
haftmann@36301
   734
*)
haftmann@36301
   735
haftmann@36301
   736
lemma divide_pos_pos:
haftmann@36301
   737
  "0 < x ==> 0 < y ==> 0 < x / y"
haftmann@36301
   738
by(simp add:field_simps)
haftmann@36301
   739
haftmann@36301
   740
lemma divide_nonneg_pos:
haftmann@36301
   741
  "0 <= x ==> 0 < y ==> 0 <= x / y"
haftmann@36301
   742
by(simp add:field_simps)
haftmann@36301
   743
haftmann@36301
   744
lemma divide_neg_pos:
haftmann@36301
   745
  "x < 0 ==> 0 < y ==> x / y < 0"
haftmann@36301
   746
by(simp add:field_simps)
haftmann@36301
   747
haftmann@36301
   748
lemma divide_nonpos_pos:
haftmann@36301
   749
  "x <= 0 ==> 0 < y ==> x / y <= 0"
haftmann@36301
   750
by(simp add:field_simps)
haftmann@36301
   751
haftmann@36301
   752
lemma divide_pos_neg:
haftmann@36301
   753
  "0 < x ==> y < 0 ==> x / y < 0"
haftmann@36301
   754
by(simp add:field_simps)
haftmann@36301
   755
haftmann@36301
   756
lemma divide_nonneg_neg:
haftmann@36301
   757
  "0 <= x ==> y < 0 ==> x / y <= 0" 
haftmann@36301
   758
by(simp add:field_simps)
haftmann@36301
   759
haftmann@36301
   760
lemma divide_neg_neg:
haftmann@36301
   761
  "x < 0 ==> y < 0 ==> 0 < x / y"
haftmann@36301
   762
by(simp add:field_simps)
haftmann@36301
   763
haftmann@36301
   764
lemma divide_nonpos_neg:
haftmann@36301
   765
  "x <= 0 ==> y < 0 ==> 0 <= x / y"
haftmann@36301
   766
by(simp add:field_simps)
haftmann@36301
   767
haftmann@36301
   768
lemma divide_strict_right_mono:
haftmann@36301
   769
     "[|a < b; 0 < c|] ==> a / c < b / c"
haftmann@36301
   770
by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono 
haftmann@36301
   771
              positive_imp_inverse_positive)
haftmann@36301
   772
haftmann@36301
   773
haftmann@36301
   774
lemma divide_strict_right_mono_neg:
haftmann@36301
   775
     "[|b < a; c < 0|] ==> a / c < b / c"
haftmann@36301
   776
apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
haftmann@36301
   777
apply (simp add: less_imp_not_eq nonzero_minus_divide_right [symmetric])
haftmann@36301
   778
done
haftmann@36301
   779
haftmann@36301
   780
text{*The last premise ensures that @{term a} and @{term b} 
haftmann@36301
   781
      have the same sign*}
haftmann@36301
   782
lemma divide_strict_left_mono:
haftmann@36301
   783
  "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / b"
huffman@44921
   784
  by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono)
haftmann@36301
   785
haftmann@36301
   786
lemma divide_left_mono:
haftmann@36301
   787
  "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / b"
huffman@44921
   788
  by (auto simp: field_simps zero_less_mult_iff mult_right_mono)
haftmann@36301
   789
haftmann@36301
   790
lemma divide_strict_left_mono_neg:
haftmann@36301
   791
  "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / b"
huffman@44921
   792
  by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono_neg)
haftmann@36301
   793
haftmann@36301
   794
lemma mult_imp_div_pos_le: "0 < y ==> x <= z * y ==>
haftmann@36301
   795
    x / y <= z"
haftmann@36301
   796
by (subst pos_divide_le_eq, assumption+)
haftmann@36301
   797
haftmann@36301
   798
lemma mult_imp_le_div_pos: "0 < y ==> z * y <= x ==>
haftmann@36301
   799
    z <= x / y"
haftmann@36301
   800
by(simp add:field_simps)
haftmann@36301
   801
haftmann@36301
   802
lemma mult_imp_div_pos_less: "0 < y ==> x < z * y ==>
haftmann@36301
   803
    x / y < z"
haftmann@36301
   804
by(simp add:field_simps)
haftmann@36301
   805
haftmann@36301
   806
lemma mult_imp_less_div_pos: "0 < y ==> z * y < x ==>
haftmann@36301
   807
    z < x / y"
haftmann@36301
   808
by(simp add:field_simps)
haftmann@36301
   809
haftmann@36301
   810
lemma frac_le: "0 <= x ==> 
haftmann@36301
   811
    x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
haftmann@36301
   812
  apply (rule mult_imp_div_pos_le)
haftmann@36301
   813
  apply simp
haftmann@36301
   814
  apply (subst times_divide_eq_left)
haftmann@36301
   815
  apply (rule mult_imp_le_div_pos, assumption)
haftmann@36301
   816
  apply (rule mult_mono)
haftmann@36301
   817
  apply simp_all
haftmann@36301
   818
done
haftmann@36301
   819
haftmann@36301
   820
lemma frac_less: "0 <= x ==> 
haftmann@36301
   821
    x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
haftmann@36301
   822
  apply (rule mult_imp_div_pos_less)
haftmann@36301
   823
  apply simp
haftmann@36301
   824
  apply (subst times_divide_eq_left)
haftmann@36301
   825
  apply (rule mult_imp_less_div_pos, assumption)
haftmann@36301
   826
  apply (erule mult_less_le_imp_less)
haftmann@36301
   827
  apply simp_all
haftmann@36301
   828
done
haftmann@36301
   829
haftmann@36301
   830
lemma frac_less2: "0 < x ==> 
haftmann@36301
   831
    x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
haftmann@36301
   832
  apply (rule mult_imp_div_pos_less)
haftmann@36301
   833
  apply simp_all
haftmann@36301
   834
  apply (rule mult_imp_less_div_pos, assumption)
haftmann@36301
   835
  apply (erule mult_le_less_imp_less)
haftmann@36301
   836
  apply simp_all
haftmann@36301
   837
done
haftmann@36301
   838
haftmann@36301
   839
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1)"
haftmann@36301
   840
by (simp add: field_simps zero_less_two)
haftmann@36301
   841
haftmann@36301
   842
lemma gt_half_sum: "a < b ==> (a+b)/(1+1) < b"
haftmann@36301
   843
by (simp add: field_simps zero_less_two)
haftmann@36301
   844
haftmann@36301
   845
subclass dense_linorder
haftmann@36301
   846
proof
haftmann@36301
   847
  fix x y :: 'a
haftmann@36301
   848
  from less_add_one show "\<exists>y. x < y" .. 
haftmann@36301
   849
  from less_add_one have "x + (- 1) < (x + 1) + (- 1)" by (rule add_strict_right_mono)
haftmann@36301
   850
  then have "x - 1 < x + 1 - 1" by (simp only: diff_minus [symmetric])
haftmann@36301
   851
  then have "x - 1 < x" by (simp add: algebra_simps)
haftmann@36301
   852
  then show "\<exists>y. y < x" ..
haftmann@36301
   853
  show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
haftmann@36301
   854
qed
haftmann@36301
   855
haftmann@36301
   856
lemma nonzero_abs_inverse:
haftmann@36301
   857
     "a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
haftmann@36301
   858
apply (auto simp add: neq_iff abs_if nonzero_inverse_minus_eq 
haftmann@36301
   859
                      negative_imp_inverse_negative)
haftmann@36301
   860
apply (blast intro: positive_imp_inverse_positive elim: less_asym) 
haftmann@36301
   861
done
haftmann@36301
   862
haftmann@36301
   863
lemma nonzero_abs_divide:
haftmann@36301
   864
     "b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
haftmann@36301
   865
  by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
haftmann@36301
   866
haftmann@36301
   867
lemma field_le_epsilon:
haftmann@36301
   868
  assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
haftmann@36301
   869
  shows "x \<le> y"
haftmann@36301
   870
proof (rule dense_le)
haftmann@36301
   871
  fix t assume "t < x"
haftmann@36301
   872
  hence "0 < x - t" by (simp add: less_diff_eq)
haftmann@36301
   873
  from e [OF this] have "x + 0 \<le> x + (y - t)" by (simp add: algebra_simps)
haftmann@36301
   874
  then have "0 \<le> y - t" by (simp only: add_le_cancel_left)
haftmann@36301
   875
  then show "t \<le> y" by (simp add: algebra_simps)
haftmann@36301
   876
qed
haftmann@36301
   877
haftmann@36301
   878
end
haftmann@36301
   879
haftmann@36414
   880
class linordered_field_inverse_zero = linordered_field + field_inverse_zero
haftmann@36348
   881
begin
haftmann@36348
   882
haftmann@36301
   883
lemma le_divide_eq:
haftmann@36301
   884
  "(a \<le> b/c) = 
haftmann@36301
   885
   (if 0 < c then a*c \<le> b
haftmann@36301
   886
             else if c < 0 then b \<le> a*c
haftmann@36409
   887
             else  a \<le> 0)"
haftmann@36301
   888
apply (cases "c=0", simp) 
haftmann@36301
   889
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
haftmann@36301
   890
done
haftmann@36301
   891
paulson@14277
   892
lemma inverse_positive_iff_positive [simp]:
haftmann@36409
   893
  "(0 < inverse a) = (0 < a)"
haftmann@21328
   894
apply (cases "a = 0", simp)
paulson@14277
   895
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
paulson@14277
   896
done
paulson@14277
   897
paulson@14277
   898
lemma inverse_negative_iff_negative [simp]:
haftmann@36409
   899
  "(inverse a < 0) = (a < 0)"
haftmann@21328
   900
apply (cases "a = 0", simp)
paulson@14277
   901
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
paulson@14277
   902
done
paulson@14277
   903
paulson@14277
   904
lemma inverse_nonnegative_iff_nonnegative [simp]:
haftmann@36409
   905
  "0 \<le> inverse a \<longleftrightarrow> 0 \<le> a"
haftmann@36409
   906
  by (simp add: not_less [symmetric])
paulson@14277
   907
paulson@14277
   908
lemma inverse_nonpositive_iff_nonpositive [simp]:
haftmann@36409
   909
  "inverse a \<le> 0 \<longleftrightarrow> a \<le> 0"
haftmann@36409
   910
  by (simp add: not_less [symmetric])
paulson@14277
   911
paulson@14365
   912
lemma one_less_inverse_iff:
haftmann@36409
   913
  "1 < inverse x \<longleftrightarrow> 0 < x \<and> x < 1"
nipkow@23482
   914
proof cases
paulson@14365
   915
  assume "0 < x"
paulson@14365
   916
    with inverse_less_iff_less [OF zero_less_one, of x]
paulson@14365
   917
    show ?thesis by simp
paulson@14365
   918
next
paulson@14365
   919
  assume notless: "~ (0 < x)"
paulson@14365
   920
  have "~ (1 < inverse x)"
paulson@14365
   921
  proof
paulson@14365
   922
    assume "1 < inverse x"
haftmann@36409
   923
    also with notless have "... \<le> 0" by simp
paulson@14365
   924
    also have "... < 1" by (rule zero_less_one) 
paulson@14365
   925
    finally show False by auto
paulson@14365
   926
  qed
paulson@14365
   927
  with notless show ?thesis by simp
paulson@14365
   928
qed
paulson@14365
   929
paulson@14365
   930
lemma one_le_inverse_iff:
haftmann@36409
   931
  "1 \<le> inverse x \<longleftrightarrow> 0 < x \<and> x \<le> 1"
haftmann@36409
   932
proof (cases "x = 1")
haftmann@36409
   933
  case True then show ?thesis by simp
haftmann@36409
   934
next
haftmann@36409
   935
  case False then have "inverse x \<noteq> 1" by simp
haftmann@36409
   936
  then have "1 \<noteq> inverse x" by blast
haftmann@36409
   937
  then have "1 \<le> inverse x \<longleftrightarrow> 1 < inverse x" by (simp add: le_less)
haftmann@36409
   938
  with False show ?thesis by (auto simp add: one_less_inverse_iff)
haftmann@36409
   939
qed
paulson@14365
   940
paulson@14365
   941
lemma inverse_less_1_iff:
haftmann@36409
   942
  "inverse x < 1 \<longleftrightarrow> x \<le> 0 \<or> 1 < x"
haftmann@36409
   943
  by (simp add: not_le [symmetric] one_le_inverse_iff) 
paulson@14365
   944
paulson@14365
   945
lemma inverse_le_1_iff:
haftmann@36409
   946
  "inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x"
haftmann@36409
   947
  by (simp add: not_less [symmetric] one_less_inverse_iff) 
paulson@14365
   948
paulson@14288
   949
lemma divide_le_eq:
paulson@14288
   950
  "(b/c \<le> a) = 
paulson@14288
   951
   (if 0 < c then b \<le> a*c
paulson@14288
   952
             else if c < 0 then a*c \<le> b
haftmann@36409
   953
             else 0 \<le> a)"
haftmann@21328
   954
apply (cases "c=0", simp) 
haftmann@36409
   955
apply (force simp add: pos_divide_le_eq neg_divide_le_eq) 
paulson@14288
   956
done
paulson@14288
   957
paulson@14288
   958
lemma less_divide_eq:
paulson@14288
   959
  "(a < b/c) = 
paulson@14288
   960
   (if 0 < c then a*c < b
paulson@14288
   961
             else if c < 0 then b < a*c
haftmann@36409
   962
             else  a < 0)"
haftmann@21328
   963
apply (cases "c=0", simp) 
haftmann@36409
   964
apply (force simp add: pos_less_divide_eq neg_less_divide_eq) 
paulson@14288
   965
done
paulson@14288
   966
paulson@14288
   967
lemma divide_less_eq:
paulson@14288
   968
  "(b/c < a) = 
paulson@14288
   969
   (if 0 < c then b < a*c
paulson@14288
   970
             else if c < 0 then a*c < b
haftmann@36409
   971
             else 0 < a)"
haftmann@21328
   972
apply (cases "c=0", simp) 
haftmann@36409
   973
apply (force simp add: pos_divide_less_eq neg_divide_less_eq)
paulson@14288
   974
done
paulson@14288
   975
haftmann@36301
   976
text {*Division and Signs*}
avigad@16775
   977
avigad@16775
   978
lemma zero_less_divide_iff:
haftmann@36409
   979
     "(0 < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
avigad@16775
   980
by (simp add: divide_inverse zero_less_mult_iff)
avigad@16775
   981
avigad@16775
   982
lemma divide_less_0_iff:
haftmann@36409
   983
     "(a/b < 0) = 
avigad@16775
   984
      (0 < a & b < 0 | a < 0 & 0 < b)"
avigad@16775
   985
by (simp add: divide_inverse mult_less_0_iff)
avigad@16775
   986
avigad@16775
   987
lemma zero_le_divide_iff:
haftmann@36409
   988
     "(0 \<le> a/b) =
avigad@16775
   989
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
avigad@16775
   990
by (simp add: divide_inverse zero_le_mult_iff)
avigad@16775
   991
avigad@16775
   992
lemma divide_le_0_iff:
haftmann@36409
   993
     "(a/b \<le> 0) =
avigad@16775
   994
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
avigad@16775
   995
by (simp add: divide_inverse mult_le_0_iff)
avigad@16775
   996
haftmann@36301
   997
text {* Division and the Number One *}
paulson@14353
   998
paulson@14353
   999
text{*Simplify expressions equated with 1*}
paulson@14353
  1000
blanchet@35828
  1001
lemma zero_eq_1_divide_iff [simp,no_atp]:
haftmann@36409
  1002
     "(0 = 1/a) = (a = 0)"
nipkow@23482
  1003
apply (cases "a=0", simp)
nipkow@23482
  1004
apply (auto simp add: nonzero_eq_divide_eq)
paulson@14353
  1005
done
paulson@14353
  1006
blanchet@35828
  1007
lemma one_divide_eq_0_iff [simp,no_atp]:
haftmann@36409
  1008
     "(1/a = 0) = (a = 0)"
nipkow@23482
  1009
apply (cases "a=0", simp)
nipkow@23482
  1010
apply (insert zero_neq_one [THEN not_sym])
nipkow@23482
  1011
apply (auto simp add: nonzero_divide_eq_eq)
paulson@14353
  1012
done
paulson@14353
  1013
paulson@14353
  1014
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
haftmann@36423
  1015
haftmann@36423
  1016
lemma zero_le_divide_1_iff [simp, no_atp]:
haftmann@36423
  1017
  "0 \<le> 1 / a \<longleftrightarrow> 0 \<le> a"
haftmann@36423
  1018
  by (simp add: zero_le_divide_iff)
paulson@17085
  1019
haftmann@36423
  1020
lemma zero_less_divide_1_iff [simp, no_atp]:
haftmann@36423
  1021
  "0 < 1 / a \<longleftrightarrow> 0 < a"
haftmann@36423
  1022
  by (simp add: zero_less_divide_iff)
haftmann@36423
  1023
haftmann@36423
  1024
lemma divide_le_0_1_iff [simp, no_atp]:
haftmann@36423
  1025
  "1 / a \<le> 0 \<longleftrightarrow> a \<le> 0"
haftmann@36423
  1026
  by (simp add: divide_le_0_iff)
haftmann@36423
  1027
haftmann@36423
  1028
lemma divide_less_0_1_iff [simp, no_atp]:
haftmann@36423
  1029
  "1 / a < 0 \<longleftrightarrow> a < 0"
haftmann@36423
  1030
  by (simp add: divide_less_0_iff)
paulson@14353
  1031
paulson@14293
  1032
lemma divide_right_mono:
haftmann@36409
  1033
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/c"
haftmann@36409
  1034
by (force simp add: divide_strict_right_mono le_less)
paulson@14293
  1035
haftmann@36409
  1036
lemma divide_right_mono_neg: "a <= b 
avigad@16775
  1037
    ==> c <= 0 ==> b / c <= a / c"
nipkow@23482
  1038
apply (drule divide_right_mono [of _ _ "- c"])
nipkow@23482
  1039
apply auto
avigad@16775
  1040
done
avigad@16775
  1041
haftmann@36409
  1042
lemma divide_left_mono_neg: "a <= b 
avigad@16775
  1043
    ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
avigad@16775
  1044
  apply (drule divide_left_mono [of _ _ "- c"])
avigad@16775
  1045
  apply (auto simp add: mult_commute)
avigad@16775
  1046
done
avigad@16775
  1047
hoelzl@42904
  1048
lemma inverse_le_iff:
hoelzl@42904
  1049
  "inverse a \<le> inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b \<le> a) \<and> (a * b \<le> 0 \<longrightarrow> a \<le> b)"
hoelzl@42904
  1050
proof -
hoelzl@42904
  1051
  { assume "a < 0"
hoelzl@42904
  1052
    then have "inverse a < 0" by simp
hoelzl@42904
  1053
    moreover assume "0 < b"
hoelzl@42904
  1054
    then have "0 < inverse b" by simp
hoelzl@42904
  1055
    ultimately have "inverse a < inverse b" by (rule less_trans)
hoelzl@42904
  1056
    then have "inverse a \<le> inverse b" by simp }
hoelzl@42904
  1057
  moreover
hoelzl@42904
  1058
  { assume "b < 0"
hoelzl@42904
  1059
    then have "inverse b < 0" by simp
hoelzl@42904
  1060
    moreover assume "0 < a"
hoelzl@42904
  1061
    then have "0 < inverse a" by simp
hoelzl@42904
  1062
    ultimately have "inverse b < inverse a" by (rule less_trans)
hoelzl@42904
  1063
    then have "\<not> inverse a \<le> inverse b" by simp }
hoelzl@42904
  1064
  ultimately show ?thesis
hoelzl@42904
  1065
    by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
hoelzl@42904
  1066
       (auto simp: not_less zero_less_mult_iff mult_le_0_iff)
hoelzl@42904
  1067
qed
hoelzl@42904
  1068
hoelzl@42904
  1069
lemma inverse_less_iff:
hoelzl@42904
  1070
  "inverse a < inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b < a) \<and> (a * b \<le> 0 \<longrightarrow> a < b)"
hoelzl@42904
  1071
  by (subst less_le) (auto simp: inverse_le_iff)
hoelzl@42904
  1072
hoelzl@42904
  1073
lemma divide_le_cancel:
hoelzl@42904
  1074
  "a / c \<le> b / c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
hoelzl@42904
  1075
  by (simp add: divide_inverse mult_le_cancel_right)
hoelzl@42904
  1076
hoelzl@42904
  1077
lemma divide_less_cancel:
hoelzl@42904
  1078
  "a / c < b / c \<longleftrightarrow> (0 < c \<longrightarrow> a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0"
hoelzl@42904
  1079
  by (auto simp add: divide_inverse mult_less_cancel_right)
hoelzl@42904
  1080
avigad@16775
  1081
text{*Simplify quotients that are compared with the value 1.*}
avigad@16775
  1082
blanchet@35828
  1083
lemma le_divide_eq_1 [no_atp]:
haftmann@36409
  1084
  "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
avigad@16775
  1085
by (auto simp add: le_divide_eq)
avigad@16775
  1086
blanchet@35828
  1087
lemma divide_le_eq_1 [no_atp]:
haftmann@36409
  1088
  "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
avigad@16775
  1089
by (auto simp add: divide_le_eq)
avigad@16775
  1090
blanchet@35828
  1091
lemma less_divide_eq_1 [no_atp]:
haftmann@36409
  1092
  "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
avigad@16775
  1093
by (auto simp add: less_divide_eq)
avigad@16775
  1094
blanchet@35828
  1095
lemma divide_less_eq_1 [no_atp]:
haftmann@36409
  1096
  "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
avigad@16775
  1097
by (auto simp add: divide_less_eq)
avigad@16775
  1098
wenzelm@23389
  1099
haftmann@36301
  1100
text {*Conditional Simplification Rules: No Case Splits*}
avigad@16775
  1101
blanchet@35828
  1102
lemma le_divide_eq_1_pos [simp,no_atp]:
haftmann@36409
  1103
  "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
avigad@16775
  1104
by (auto simp add: le_divide_eq)
avigad@16775
  1105
blanchet@35828
  1106
lemma le_divide_eq_1_neg [simp,no_atp]:
haftmann@36409
  1107
  "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
avigad@16775
  1108
by (auto simp add: le_divide_eq)
avigad@16775
  1109
blanchet@35828
  1110
lemma divide_le_eq_1_pos [simp,no_atp]:
haftmann@36409
  1111
  "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
avigad@16775
  1112
by (auto simp add: divide_le_eq)
avigad@16775
  1113
blanchet@35828
  1114
lemma divide_le_eq_1_neg [simp,no_atp]:
haftmann@36409
  1115
  "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
avigad@16775
  1116
by (auto simp add: divide_le_eq)
avigad@16775
  1117
blanchet@35828
  1118
lemma less_divide_eq_1_pos [simp,no_atp]:
haftmann@36409
  1119
  "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
avigad@16775
  1120
by (auto simp add: less_divide_eq)
avigad@16775
  1121
blanchet@35828
  1122
lemma less_divide_eq_1_neg [simp,no_atp]:
haftmann@36409
  1123
  "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
avigad@16775
  1124
by (auto simp add: less_divide_eq)
avigad@16775
  1125
blanchet@35828
  1126
lemma divide_less_eq_1_pos [simp,no_atp]:
haftmann@36409
  1127
  "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
paulson@18649
  1128
by (auto simp add: divide_less_eq)
paulson@18649
  1129
blanchet@35828
  1130
lemma divide_less_eq_1_neg [simp,no_atp]:
haftmann@36409
  1131
  "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
avigad@16775
  1132
by (auto simp add: divide_less_eq)
avigad@16775
  1133
blanchet@35828
  1134
lemma eq_divide_eq_1 [simp,no_atp]:
haftmann@36409
  1135
  "(1 = b/a) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1136
by (auto simp add: eq_divide_eq)
avigad@16775
  1137
blanchet@35828
  1138
lemma divide_eq_eq_1 [simp,no_atp]:
haftmann@36409
  1139
  "(b/a = 1) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1140
by (auto simp add: divide_eq_eq)
avigad@16775
  1141
paulson@14294
  1142
lemma abs_inverse [simp]:
haftmann@36409
  1143
     "\<bar>inverse a\<bar> = 
haftmann@36301
  1144
      inverse \<bar>a\<bar>"
haftmann@21328
  1145
apply (cases "a=0", simp) 
paulson@14294
  1146
apply (simp add: nonzero_abs_inverse) 
paulson@14294
  1147
done
paulson@14294
  1148
paulson@15234
  1149
lemma abs_divide [simp]:
haftmann@36409
  1150
     "\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
haftmann@21328
  1151
apply (cases "b=0", simp) 
paulson@14294
  1152
apply (simp add: nonzero_abs_divide) 
paulson@14294
  1153
done
paulson@14294
  1154
haftmann@36409
  1155
lemma abs_div_pos: "0 < y ==> 
haftmann@36301
  1156
    \<bar>x\<bar> / y = \<bar>x / y\<bar>"
haftmann@25304
  1157
  apply (subst abs_divide)
haftmann@25304
  1158
  apply (simp add: order_less_imp_le)
haftmann@25304
  1159
done
avigad@16775
  1160
hoelzl@35579
  1161
lemma field_le_mult_one_interval:
hoelzl@35579
  1162
  assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
hoelzl@35579
  1163
  shows "x \<le> y"
hoelzl@35579
  1164
proof (cases "0 < x")
hoelzl@35579
  1165
  assume "0 < x"
hoelzl@35579
  1166
  thus ?thesis
hoelzl@35579
  1167
    using dense_le_bounded[of 0 1 "y/x"] *
hoelzl@35579
  1168
    unfolding le_divide_eq if_P[OF `0 < x`] by simp
hoelzl@35579
  1169
next
hoelzl@35579
  1170
  assume "\<not>0 < x" hence "x \<le> 0" by simp
hoelzl@35579
  1171
  obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1\<Colon>'a"] by auto
hoelzl@35579
  1172
  hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] `x \<le> 0` by auto
hoelzl@35579
  1173
  also note *[OF s]
hoelzl@35579
  1174
  finally show ?thesis .
hoelzl@35579
  1175
qed
haftmann@35090
  1176
haftmann@36409
  1177
end
haftmann@36409
  1178
haftmann@33364
  1179
code_modulename SML
haftmann@35050
  1180
  Fields Arith
haftmann@33364
  1181
haftmann@33364
  1182
code_modulename OCaml
haftmann@35050
  1183
  Fields Arith
haftmann@33364
  1184
haftmann@33364
  1185
code_modulename Haskell
haftmann@35050
  1186
  Fields Arith
haftmann@33364
  1187
paulson@14265
  1188
end