src/HOL/Fields.thy
author hoelzl
Wed Apr 09 09:37:47 2014 +0200 (2014-04-09)
changeset 56479 91958d4b30f7
parent 56445 82ce19612fe8
child 56480 093ea91498e6
permissions -rw-r--r--
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
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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]: "(-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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  using add_divide_distrib [of a "- b" c] by simp
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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 nonzero_neg_divide_eq_eq[field_simps]:
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  "b \<noteq> 0 \<Longrightarrow> -(a/b) = c \<longleftrightarrow> -a = c*b"
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using nonzero_divide_eq_eq[of b "-a" c] by (simp add: divide_minus_left)
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lemma nonzero_neg_divide_eq_eq2[field_simps]:
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  "b \<noteq> 0 \<Longrightarrow> c = -(a/b) \<longleftrightarrow> c*b = -a"
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using nonzero_neg_divide_eq_eq[of b a c] by auto
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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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lemma add_divide_eq_iff [field_simps]:
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  "z \<noteq> 0 \<Longrightarrow> x + y / z = (x * z + y) / z"
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  by (simp add: add_divide_distrib nonzero_eq_divide_eq)
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lemma divide_add_eq_iff [field_simps]:
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  "z \<noteq> 0 \<Longrightarrow> x / z + y = (x + y * z) / z"
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  by (simp add: add_divide_distrib nonzero_eq_divide_eq)
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lemma diff_divide_eq_iff [field_simps]:
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  "z \<noteq> 0 \<Longrightarrow> x - y / z = (x * z - y) / z"
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  by (simp add: diff_divide_distrib nonzero_eq_divide_eq eq_diff_eq)
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lemma divide_diff_eq_iff [field_simps]:
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  "z \<noteq> 0 \<Longrightarrow> x / z - y = (x - y * z) / z"
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  by (simp add: field_simps)
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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]:
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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]:
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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 = 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]:
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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]:
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  "a \<noteq> 0 \<Longrightarrow> a * b / a = b"
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   324
using nonzero_mult_divide_mult_cancel_left [of 1 a b] by simp
huffman@30630
   325
blanchet@54147
   326
lemma nonzero_divide_mult_cancel_right [simp]:
huffman@30630
   327
  "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> b / (a * b) = 1 / a"
huffman@30630
   328
using nonzero_mult_divide_mult_cancel_right [of a b 1] by simp
huffman@30630
   329
blanchet@54147
   330
lemma nonzero_divide_mult_cancel_left [simp]:
huffman@30630
   331
  "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / (a * b) = 1 / b"
huffman@30630
   332
using nonzero_mult_divide_mult_cancel_left [of b a 1] by simp
huffman@30630
   333
blanchet@54147
   334
lemma nonzero_mult_divide_mult_cancel_left2 [simp]:
huffman@30630
   335
  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (c * a) / (b * c) = a / b"
huffman@30630
   336
using nonzero_mult_divide_mult_cancel_left [of b c a] by (simp add: mult_ac)
huffman@30630
   337
blanchet@54147
   338
lemma nonzero_mult_divide_mult_cancel_right2 [simp]:
huffman@30630
   339
  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (c * b) = a / b"
huffman@30630
   340
using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: mult_ac)
huffman@30630
   341
huffman@30630
   342
lemma diff_frac_eq:
huffman@30630
   343
  "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y - w / z = (x * z - w * y) / (y * z)"
haftmann@36348
   344
  by (simp add: field_simps)
huffman@30630
   345
huffman@30630
   346
lemma frac_eq_eq:
huffman@30630
   347
  "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)"
haftmann@36348
   348
  by (simp add: field_simps)
haftmann@36348
   349
haftmann@36348
   350
end
haftmann@36348
   351
haftmann@36348
   352
class field_inverse_zero = field +
haftmann@36348
   353
  assumes field_inverse_zero: "inverse 0 = 0"
haftmann@36348
   354
begin
haftmann@36348
   355
haftmann@36348
   356
subclass division_ring_inverse_zero proof
haftmann@36348
   357
qed (fact field_inverse_zero)
haftmann@25230
   358
paulson@14270
   359
text{*This version builds in division by zero while also re-orienting
paulson@14270
   360
      the right-hand side.*}
paulson@14270
   361
lemma inverse_mult_distrib [simp]:
haftmann@36409
   362
  "inverse (a * b) = inverse a * inverse b"
haftmann@36409
   363
proof cases
haftmann@36409
   364
  assume "a \<noteq> 0 & b \<noteq> 0" 
haftmann@36409
   365
  thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_ac)
haftmann@36409
   366
next
haftmann@36409
   367
  assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
haftmann@36409
   368
  thus ?thesis by force
haftmann@36409
   369
qed
paulson@14270
   370
paulson@14365
   371
lemma inverse_divide [simp]:
haftmann@36409
   372
  "inverse (a / b) = b / a"
haftmann@36301
   373
  by (simp add: divide_inverse mult_commute)
paulson@14365
   374
wenzelm@23389
   375
haftmann@36301
   376
text {* Calculations with fractions *}
avigad@16775
   377
nipkow@23413
   378
text{* There is a whole bunch of simp-rules just for class @{text
nipkow@23413
   379
field} but none for class @{text field} and @{text nonzero_divides}
nipkow@23413
   380
because the latter are covered by a simproc. *}
nipkow@23413
   381
nipkow@23413
   382
lemma mult_divide_mult_cancel_left:
haftmann@36409
   383
  "c \<noteq> 0 \<Longrightarrow> (c * a) / (c * b) = a / b"
haftmann@21328
   384
apply (cases "b = 0")
huffman@35216
   385
apply simp_all
paulson@14277
   386
done
paulson@14277
   387
nipkow@23413
   388
lemma mult_divide_mult_cancel_right:
haftmann@36409
   389
  "c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b"
haftmann@21328
   390
apply (cases "b = 0")
huffman@35216
   391
apply simp_all
paulson@14321
   392
done
nipkow@23413
   393
blanchet@54147
   394
lemma divide_divide_eq_right [simp]:
haftmann@36409
   395
  "a / (b / c) = (a * c) / b"
haftmann@36409
   396
  by (simp add: divide_inverse mult_ac)
paulson@14288
   397
blanchet@54147
   398
lemma divide_divide_eq_left [simp]:
haftmann@36409
   399
  "(a / b) / c = a / (b * c)"
haftmann@36409
   400
  by (simp add: divide_inverse mult_assoc)
paulson@14288
   401
lp15@56365
   402
lemma divide_divide_times_eq:
lp15@56365
   403
  "(x / y) / (z / w) = (x * w) / (y * z)"
lp15@56365
   404
  by simp
wenzelm@23389
   405
haftmann@36301
   406
text {*Special Cancellation Simprules for Division*}
paulson@15234
   407
blanchet@54147
   408
lemma mult_divide_mult_cancel_left_if [simp]:
haftmann@36409
   409
  shows "(c * a) / (c * b) = (if c = 0 then 0 else a / b)"
haftmann@36409
   410
  by (simp add: mult_divide_mult_cancel_left)
nipkow@23413
   411
paulson@15234
   412
haftmann@36301
   413
text {* Division and Unary Minus *}
paulson@14293
   414
haftmann@36409
   415
lemma minus_divide_right:
haftmann@36409
   416
  "- (a / b) = a / - b"
haftmann@36409
   417
  by (simp add: divide_inverse)
paulson@14430
   418
hoelzl@56479
   419
lemma divide_minus_right [simp]:
haftmann@36409
   420
  "a / - b = - (a / b)"
haftmann@36409
   421
  by (simp add: divide_inverse)
huffman@30630
   422
hoelzl@56479
   423
lemma minus_divide_divide:
haftmann@36409
   424
  "(- a) / (- b) = a / b"
haftmann@21328
   425
apply (cases "b=0", simp) 
paulson@14293
   426
apply (simp add: nonzero_minus_divide_divide) 
paulson@14293
   427
done
paulson@14293
   428
nipkow@23482
   429
lemma eq_divide_eq:
haftmann@36409
   430
  "a = b / c \<longleftrightarrow> (if c \<noteq> 0 then a * c = b else a = 0)"
haftmann@36409
   431
  by (simp add: nonzero_eq_divide_eq)
nipkow@23482
   432
nipkow@23482
   433
lemma divide_eq_eq:
haftmann@36409
   434
  "b / c = a \<longleftrightarrow> (if c \<noteq> 0 then b = a * c else a = 0)"
haftmann@36409
   435
  by (force simp add: nonzero_divide_eq_eq)
paulson@14293
   436
haftmann@36301
   437
lemma inverse_eq_1_iff [simp]:
haftmann@36409
   438
  "inverse x = 1 \<longleftrightarrow> x = 1"
haftmann@36409
   439
  by (insert inverse_eq_iff_eq [of x 1], simp) 
wenzelm@23389
   440
blanchet@54147
   441
lemma divide_eq_0_iff [simp]:
haftmann@36409
   442
  "a / b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
haftmann@36409
   443
  by (simp add: divide_inverse)
haftmann@36301
   444
blanchet@54147
   445
lemma divide_cancel_right [simp]:
haftmann@36409
   446
  "a / c = b / c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@36409
   447
  apply (cases "c=0", simp)
haftmann@36409
   448
  apply (simp add: divide_inverse)
haftmann@36409
   449
  done
haftmann@36301
   450
blanchet@54147
   451
lemma divide_cancel_left [simp]:
haftmann@36409
   452
  "c / a = c / b \<longleftrightarrow> c = 0 \<or> a = b" 
haftmann@36409
   453
  apply (cases "c=0", simp)
haftmann@36409
   454
  apply (simp add: divide_inverse)
haftmann@36409
   455
  done
haftmann@36301
   456
blanchet@54147
   457
lemma divide_eq_1_iff [simp]:
haftmann@36409
   458
  "a / b = 1 \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
haftmann@36409
   459
  apply (cases "b=0", simp)
haftmann@36409
   460
  apply (simp add: right_inverse_eq)
haftmann@36409
   461
  done
haftmann@36301
   462
blanchet@54147
   463
lemma one_eq_divide_iff [simp]:
haftmann@36409
   464
  "1 = a / b \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
haftmann@36409
   465
  by (simp add: eq_commute [of 1])
haftmann@36409
   466
haftmann@36719
   467
lemma times_divide_times_eq:
haftmann@36719
   468
  "(x / y) * (z / w) = (x * z) / (y * w)"
haftmann@36719
   469
  by simp
haftmann@36719
   470
haftmann@36719
   471
lemma add_frac_num:
haftmann@36719
   472
  "y \<noteq> 0 \<Longrightarrow> x / y + z = (x + z * y) / y"
haftmann@36719
   473
  by (simp add: add_divide_distrib)
haftmann@36719
   474
haftmann@36719
   475
lemma add_num_frac:
haftmann@36719
   476
  "y \<noteq> 0 \<Longrightarrow> z + x / y = (x + z * y) / y"
haftmann@36719
   477
  by (simp add: add_divide_distrib add.commute)
haftmann@36719
   478
haftmann@36409
   479
end
haftmann@36301
   480
haftmann@36301
   481
huffman@44064
   482
subsection {* Ordered fields *}
haftmann@36301
   483
haftmann@36301
   484
class linordered_field = field + linordered_idom
haftmann@36301
   485
begin
paulson@14268
   486
paulson@14277
   487
lemma positive_imp_inverse_positive: 
haftmann@36301
   488
  assumes a_gt_0: "0 < a" 
haftmann@36301
   489
  shows "0 < inverse a"
nipkow@23482
   490
proof -
paulson@14268
   491
  have "0 < a * inverse a" 
haftmann@36301
   492
    by (simp add: a_gt_0 [THEN less_imp_not_eq2])
paulson@14268
   493
  thus "0 < inverse a" 
haftmann@36301
   494
    by (simp add: a_gt_0 [THEN less_not_sym] zero_less_mult_iff)
nipkow@23482
   495
qed
paulson@14268
   496
paulson@14277
   497
lemma negative_imp_inverse_negative:
haftmann@36301
   498
  "a < 0 \<Longrightarrow> inverse a < 0"
haftmann@36301
   499
  by (insert positive_imp_inverse_positive [of "-a"], 
haftmann@36301
   500
    simp add: nonzero_inverse_minus_eq less_imp_not_eq)
paulson@14268
   501
paulson@14268
   502
lemma inverse_le_imp_le:
haftmann@36301
   503
  assumes invle: "inverse a \<le> inverse b" and apos: "0 < a"
haftmann@36301
   504
  shows "b \<le> a"
nipkow@23482
   505
proof (rule classical)
paulson@14268
   506
  assume "~ b \<le> a"
nipkow@23482
   507
  hence "a < b"  by (simp add: linorder_not_le)
haftmann@36301
   508
  hence bpos: "0 < b"  by (blast intro: apos less_trans)
paulson@14268
   509
  hence "a * inverse a \<le> a * inverse b"
haftmann@36301
   510
    by (simp add: apos invle less_imp_le mult_left_mono)
paulson@14268
   511
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
haftmann@36301
   512
    by (simp add: bpos less_imp_le mult_right_mono)
haftmann@36301
   513
  thus "b \<le> a"  by (simp add: mult_assoc apos bpos less_imp_not_eq2)
nipkow@23482
   514
qed
paulson@14268
   515
paulson@14277
   516
lemma inverse_positive_imp_positive:
haftmann@36301
   517
  assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
haftmann@36301
   518
  shows "0 < a"
wenzelm@23389
   519
proof -
paulson@14277
   520
  have "0 < inverse (inverse a)"
wenzelm@23389
   521
    using inv_gt_0 by (rule positive_imp_inverse_positive)
paulson@14277
   522
  thus "0 < a"
wenzelm@23389
   523
    using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
   524
qed
paulson@14277
   525
haftmann@36301
   526
lemma inverse_negative_imp_negative:
haftmann@36301
   527
  assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0"
haftmann@36301
   528
  shows "a < 0"
haftmann@36301
   529
proof -
haftmann@36301
   530
  have "inverse (inverse a) < 0"
haftmann@36301
   531
    using inv_less_0 by (rule negative_imp_inverse_negative)
haftmann@36301
   532
  thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
haftmann@36301
   533
qed
haftmann@36301
   534
haftmann@36301
   535
lemma linordered_field_no_lb:
haftmann@36301
   536
  "\<forall>x. \<exists>y. y < x"
haftmann@36301
   537
proof
haftmann@36301
   538
  fix x::'a
haftmann@36301
   539
  have m1: "- (1::'a) < 0" by simp
haftmann@36301
   540
  from add_strict_right_mono[OF m1, where c=x] 
haftmann@36301
   541
  have "(- 1) + x < x" by simp
haftmann@36301
   542
  thus "\<exists>y. y < x" by blast
haftmann@36301
   543
qed
haftmann@36301
   544
haftmann@36301
   545
lemma linordered_field_no_ub:
haftmann@36301
   546
  "\<forall> x. \<exists>y. y > x"
haftmann@36301
   547
proof
haftmann@36301
   548
  fix x::'a
haftmann@36301
   549
  have m1: " (1::'a) > 0" by simp
haftmann@36301
   550
  from add_strict_right_mono[OF m1, where c=x] 
haftmann@36301
   551
  have "1 + x > x" by simp
haftmann@36301
   552
  thus "\<exists>y. y > x" by blast
haftmann@36301
   553
qed
haftmann@36301
   554
haftmann@36301
   555
lemma less_imp_inverse_less:
haftmann@36301
   556
  assumes less: "a < b" and apos:  "0 < a"
haftmann@36301
   557
  shows "inverse b < inverse a"
haftmann@36301
   558
proof (rule ccontr)
haftmann@36301
   559
  assume "~ inverse b < inverse a"
haftmann@36301
   560
  hence "inverse a \<le> inverse b" by simp
haftmann@36301
   561
  hence "~ (a < b)"
haftmann@36301
   562
    by (simp add: not_less inverse_le_imp_le [OF _ apos])
haftmann@36301
   563
  thus False by (rule notE [OF _ less])
haftmann@36301
   564
qed
haftmann@36301
   565
haftmann@36301
   566
lemma inverse_less_imp_less:
haftmann@36301
   567
  "inverse a < inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b < a"
haftmann@36301
   568
apply (simp add: less_le [of "inverse a"] less_le [of "b"])
haftmann@36301
   569
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
haftmann@36301
   570
done
haftmann@36301
   571
haftmann@36301
   572
text{*Both premises are essential. Consider -1 and 1.*}
blanchet@54147
   573
lemma inverse_less_iff_less [simp]:
haftmann@36301
   574
  "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
haftmann@36301
   575
  by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
haftmann@36301
   576
haftmann@36301
   577
lemma le_imp_inverse_le:
haftmann@36301
   578
  "a \<le> b \<Longrightarrow> 0 < a \<Longrightarrow> inverse b \<le> inverse a"
haftmann@36301
   579
  by (force simp add: le_less less_imp_inverse_less)
haftmann@36301
   580
blanchet@54147
   581
lemma inverse_le_iff_le [simp]:
haftmann@36301
   582
  "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
haftmann@36301
   583
  by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
haftmann@36301
   584
haftmann@36301
   585
haftmann@36301
   586
text{*These results refer to both operands being negative.  The opposite-sign
haftmann@36301
   587
case is trivial, since inverse preserves signs.*}
haftmann@36301
   588
lemma inverse_le_imp_le_neg:
haftmann@36301
   589
  "inverse a \<le> inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a"
haftmann@36301
   590
apply (rule classical) 
haftmann@36301
   591
apply (subgoal_tac "a < 0") 
haftmann@36301
   592
 prefer 2 apply force
haftmann@36301
   593
apply (insert inverse_le_imp_le [of "-b" "-a"])
haftmann@36301
   594
apply (simp add: nonzero_inverse_minus_eq) 
haftmann@36301
   595
done
haftmann@36301
   596
haftmann@36301
   597
lemma less_imp_inverse_less_neg:
haftmann@36301
   598
   "a < b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b < inverse a"
haftmann@36301
   599
apply (subgoal_tac "a < 0") 
haftmann@36301
   600
 prefer 2 apply (blast intro: less_trans) 
haftmann@36301
   601
apply (insert less_imp_inverse_less [of "-b" "-a"])
haftmann@36301
   602
apply (simp add: nonzero_inverse_minus_eq) 
haftmann@36301
   603
done
haftmann@36301
   604
haftmann@36301
   605
lemma inverse_less_imp_less_neg:
haftmann@36301
   606
   "inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a"
haftmann@36301
   607
apply (rule classical) 
haftmann@36301
   608
apply (subgoal_tac "a < 0") 
haftmann@36301
   609
 prefer 2
haftmann@36301
   610
 apply force
haftmann@36301
   611
apply (insert inverse_less_imp_less [of "-b" "-a"])
haftmann@36301
   612
apply (simp add: nonzero_inverse_minus_eq) 
haftmann@36301
   613
done
haftmann@36301
   614
blanchet@54147
   615
lemma inverse_less_iff_less_neg [simp]:
haftmann@36301
   616
  "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
haftmann@36301
   617
apply (insert inverse_less_iff_less [of "-b" "-a"])
haftmann@36301
   618
apply (simp del: inverse_less_iff_less 
haftmann@36301
   619
            add: nonzero_inverse_minus_eq)
haftmann@36301
   620
done
haftmann@36301
   621
haftmann@36301
   622
lemma le_imp_inverse_le_neg:
haftmann@36301
   623
  "a \<le> b \<Longrightarrow> b < 0 ==> inverse b \<le> inverse a"
haftmann@36301
   624
  by (force simp add: le_less less_imp_inverse_less_neg)
haftmann@36301
   625
blanchet@54147
   626
lemma inverse_le_iff_le_neg [simp]:
haftmann@36301
   627
  "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
haftmann@36301
   628
  by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
haftmann@36301
   629
huffman@36774
   630
lemma one_less_inverse:
huffman@36774
   631
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> 1 < inverse a"
huffman@36774
   632
  using less_imp_inverse_less [of a 1, unfolded inverse_1] .
huffman@36774
   633
huffman@36774
   634
lemma one_le_inverse:
huffman@36774
   635
  "0 < a \<Longrightarrow> a \<le> 1 \<Longrightarrow> 1 \<le> inverse a"
huffman@36774
   636
  using le_imp_inverse_le [of a 1, unfolded inverse_1] .
huffman@36774
   637
haftmann@36348
   638
lemma pos_le_divide_eq [field_simps]: "0 < c ==> (a \<le> b/c) = (a*c \<le> b)"
haftmann@36301
   639
proof -
haftmann@36301
   640
  assume less: "0<c"
haftmann@36301
   641
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
haftmann@36304
   642
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   643
  also have "... = (a*c \<le> b)"
haftmann@36301
   644
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
haftmann@36301
   645
  finally show ?thesis .
haftmann@36301
   646
qed
haftmann@36301
   647
haftmann@36348
   648
lemma neg_le_divide_eq [field_simps]: "c < 0 ==> (a \<le> b/c) = (b \<le> a*c)"
haftmann@36301
   649
proof -
haftmann@36301
   650
  assume less: "c<0"
haftmann@36301
   651
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
haftmann@36304
   652
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   653
  also have "... = (b \<le> a*c)"
haftmann@36301
   654
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
haftmann@36301
   655
  finally show ?thesis .
haftmann@36301
   656
qed
haftmann@36301
   657
haftmann@36348
   658
lemma pos_less_divide_eq [field_simps]:
haftmann@36301
   659
     "0 < c ==> (a < b/c) = (a*c < b)"
haftmann@36301
   660
proof -
haftmann@36301
   661
  assume less: "0<c"
haftmann@36301
   662
  hence "(a < b/c) = (a*c < (b/c)*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 "... = (a*c < b)"
haftmann@36301
   665
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
haftmann@36301
   666
  finally show ?thesis .
haftmann@36301
   667
qed
haftmann@36301
   668
haftmann@36348
   669
lemma neg_less_divide_eq [field_simps]:
haftmann@36301
   670
 "c < 0 ==> (a < b/c) = (b < a*c)"
haftmann@36301
   671
proof -
haftmann@36301
   672
  assume less: "c<0"
haftmann@36301
   673
  hence "(a < b/c) = ((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_eq [OF less] divide_inverse mult_assoc) 
haftmann@36301
   677
  finally show ?thesis .
haftmann@36301
   678
qed
haftmann@36301
   679
haftmann@36348
   680
lemma pos_divide_less_eq [field_simps]:
haftmann@36301
   681
     "0 < c ==> (b/c < a) = (b < a*c)"
haftmann@36301
   682
proof -
haftmann@36301
   683
  assume less: "0<c"
haftmann@36301
   684
  hence "(b/c < a) = ((b/c)*c < a*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 "... = (b < a*c)"
haftmann@36301
   687
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
haftmann@36301
   688
  finally show ?thesis .
haftmann@36301
   689
qed
haftmann@36301
   690
haftmann@36348
   691
lemma neg_divide_less_eq [field_simps]:
haftmann@36301
   692
 "c < 0 ==> (b/c < a) = (a*c < b)"
haftmann@36301
   693
proof -
haftmann@36301
   694
  assume less: "c<0"
haftmann@36301
   695
  hence "(b/c < a) = (a*c < (b/c)*c)"
haftmann@36304
   696
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   697
  also have "... = (a*c < b)"
haftmann@36301
   698
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
haftmann@36301
   699
  finally show ?thesis .
haftmann@36301
   700
qed
haftmann@36301
   701
haftmann@36348
   702
lemma pos_divide_le_eq [field_simps]: "0 < c ==> (b/c \<le> a) = (b \<le> a*c)"
haftmann@36301
   703
proof -
haftmann@36301
   704
  assume less: "0<c"
haftmann@36301
   705
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
haftmann@36304
   706
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   707
  also have "... = (b \<le> a*c)"
haftmann@36301
   708
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
haftmann@36301
   709
  finally show ?thesis .
haftmann@36301
   710
qed
haftmann@36301
   711
haftmann@36348
   712
lemma neg_divide_le_eq [field_simps]: "c < 0 ==> (b/c \<le> a) = (a*c \<le> b)"
haftmann@36301
   713
proof -
haftmann@36301
   714
  assume less: "c<0"
haftmann@36301
   715
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
haftmann@36304
   716
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   717
  also have "... = (a*c \<le> b)"
haftmann@36301
   718
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
haftmann@36301
   719
  finally show ?thesis .
haftmann@36301
   720
qed
haftmann@36301
   721
lp15@56365
   722
lemma frac_less_eq:
lp15@56365
   723
  "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y < w / z \<longleftrightarrow> (x * z - w * y) / (y * z) < 0"
lp15@56365
   724
  by (subst less_iff_diff_less_0) (simp add: diff_frac_eq )
lp15@56365
   725
lp15@56365
   726
lemma frac_le_eq:
lp15@56365
   727
  "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y \<le> w / z \<longleftrightarrow> (x * z - w * y) / (y * z) \<le> 0"
lp15@56365
   728
  by (subst le_iff_diff_le_0) (simp add: diff_frac_eq )
lp15@56365
   729
haftmann@36301
   730
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
haftmann@36301
   731
of positivity/negativity needed for @{text field_simps}. Have not added @{text
haftmann@36301
   732
sign_simps} to @{text field_simps} because the former can lead to case
haftmann@36301
   733
explosions. *}
haftmann@36301
   734
blanchet@54147
   735
lemmas sign_simps = algebra_simps zero_less_mult_iff mult_less_0_iff
haftmann@36348
   736
blanchet@54147
   737
lemmas (in -) sign_simps = algebra_simps zero_less_mult_iff mult_less_0_iff
haftmann@36301
   738
haftmann@36301
   739
(* Only works once linear arithmetic is installed:
haftmann@36301
   740
text{*An example:*}
haftmann@36301
   741
lemma fixes a b c d e f :: "'a::linordered_field"
haftmann@36301
   742
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
haftmann@36301
   743
 ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
haftmann@36301
   744
 ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
haftmann@36301
   745
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
haftmann@36301
   746
 prefer 2 apply(simp add:sign_simps)
haftmann@36301
   747
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
haftmann@36301
   748
 prefer 2 apply(simp add:sign_simps)
haftmann@36301
   749
apply(simp add:field_simps)
haftmann@36301
   750
done
haftmann@36301
   751
*)
haftmann@36301
   752
haftmann@36301
   753
lemma divide_pos_pos:
haftmann@36301
   754
  "0 < x ==> 0 < y ==> 0 < x / y"
haftmann@36301
   755
by(simp add:field_simps)
haftmann@36301
   756
haftmann@36301
   757
lemma divide_nonneg_pos:
haftmann@36301
   758
  "0 <= x ==> 0 < y ==> 0 <= x / y"
haftmann@36301
   759
by(simp add:field_simps)
haftmann@36301
   760
haftmann@36301
   761
lemma divide_neg_pos:
haftmann@36301
   762
  "x < 0 ==> 0 < y ==> x / y < 0"
haftmann@36301
   763
by(simp add:field_simps)
haftmann@36301
   764
haftmann@36301
   765
lemma divide_nonpos_pos:
haftmann@36301
   766
  "x <= 0 ==> 0 < y ==> x / y <= 0"
haftmann@36301
   767
by(simp add:field_simps)
haftmann@36301
   768
haftmann@36301
   769
lemma divide_pos_neg:
haftmann@36301
   770
  "0 < x ==> y < 0 ==> x / y < 0"
haftmann@36301
   771
by(simp add:field_simps)
haftmann@36301
   772
haftmann@36301
   773
lemma divide_nonneg_neg:
haftmann@36301
   774
  "0 <= x ==> y < 0 ==> x / y <= 0" 
haftmann@36301
   775
by(simp add:field_simps)
haftmann@36301
   776
haftmann@36301
   777
lemma divide_neg_neg:
haftmann@36301
   778
  "x < 0 ==> y < 0 ==> 0 < x / y"
haftmann@36301
   779
by(simp add:field_simps)
haftmann@36301
   780
haftmann@36301
   781
lemma divide_nonpos_neg:
haftmann@36301
   782
  "x <= 0 ==> y < 0 ==> 0 <= x / y"
haftmann@36301
   783
by(simp add:field_simps)
haftmann@36301
   784
haftmann@36301
   785
lemma divide_strict_right_mono:
haftmann@36301
   786
     "[|a < b; 0 < c|] ==> a / c < b / c"
haftmann@36301
   787
by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono 
haftmann@36301
   788
              positive_imp_inverse_positive)
haftmann@36301
   789
haftmann@36301
   790
haftmann@36301
   791
lemma divide_strict_right_mono_neg:
haftmann@36301
   792
     "[|b < a; c < 0|] ==> a / c < b / c"
haftmann@36301
   793
apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
haftmann@36301
   794
apply (simp add: less_imp_not_eq nonzero_minus_divide_right [symmetric])
haftmann@36301
   795
done
haftmann@36301
   796
haftmann@36301
   797
text{*The last premise ensures that @{term a} and @{term b} 
haftmann@36301
   798
      have the same sign*}
haftmann@36301
   799
lemma divide_strict_left_mono:
haftmann@36301
   800
  "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / b"
huffman@44921
   801
  by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono)
haftmann@36301
   802
haftmann@36301
   803
lemma divide_left_mono:
haftmann@36301
   804
  "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / b"
huffman@44921
   805
  by (auto simp: field_simps zero_less_mult_iff mult_right_mono)
haftmann@36301
   806
haftmann@36301
   807
lemma divide_strict_left_mono_neg:
haftmann@36301
   808
  "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / b"
huffman@44921
   809
  by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono_neg)
haftmann@36301
   810
haftmann@36301
   811
lemma mult_imp_div_pos_le: "0 < y ==> x <= z * y ==>
haftmann@36301
   812
    x / y <= z"
haftmann@36301
   813
by (subst pos_divide_le_eq, assumption+)
haftmann@36301
   814
haftmann@36301
   815
lemma mult_imp_le_div_pos: "0 < y ==> z * y <= x ==>
haftmann@36301
   816
    z <= x / y"
haftmann@36301
   817
by(simp add:field_simps)
haftmann@36301
   818
haftmann@36301
   819
lemma mult_imp_div_pos_less: "0 < y ==> x < z * y ==>
haftmann@36301
   820
    x / y < z"
haftmann@36301
   821
by(simp add:field_simps)
haftmann@36301
   822
haftmann@36301
   823
lemma mult_imp_less_div_pos: "0 < y ==> z * y < x ==>
haftmann@36301
   824
    z < x / y"
haftmann@36301
   825
by(simp add:field_simps)
haftmann@36301
   826
haftmann@36301
   827
lemma frac_le: "0 <= x ==> 
haftmann@36301
   828
    x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
haftmann@36301
   829
  apply (rule mult_imp_div_pos_le)
haftmann@36301
   830
  apply simp
haftmann@36301
   831
  apply (subst times_divide_eq_left)
haftmann@36301
   832
  apply (rule mult_imp_le_div_pos, assumption)
haftmann@36301
   833
  apply (rule mult_mono)
haftmann@36301
   834
  apply simp_all
haftmann@36301
   835
done
haftmann@36301
   836
haftmann@36301
   837
lemma frac_less: "0 <= x ==> 
haftmann@36301
   838
    x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
haftmann@36301
   839
  apply (rule mult_imp_div_pos_less)
haftmann@36301
   840
  apply simp
haftmann@36301
   841
  apply (subst times_divide_eq_left)
haftmann@36301
   842
  apply (rule mult_imp_less_div_pos, assumption)
haftmann@36301
   843
  apply (erule mult_less_le_imp_less)
haftmann@36301
   844
  apply simp_all
haftmann@36301
   845
done
haftmann@36301
   846
haftmann@36301
   847
lemma frac_less2: "0 < x ==> 
haftmann@36301
   848
    x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
haftmann@36301
   849
  apply (rule mult_imp_div_pos_less)
haftmann@36301
   850
  apply simp_all
haftmann@36301
   851
  apply (rule mult_imp_less_div_pos, assumption)
haftmann@36301
   852
  apply (erule mult_le_less_imp_less)
haftmann@36301
   853
  apply simp_all
haftmann@36301
   854
done
haftmann@36301
   855
haftmann@36301
   856
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1)"
haftmann@36301
   857
by (simp add: field_simps zero_less_two)
haftmann@36301
   858
haftmann@36301
   859
lemma gt_half_sum: "a < b ==> (a+b)/(1+1) < b"
haftmann@36301
   860
by (simp add: field_simps zero_less_two)
haftmann@36301
   861
hoelzl@53215
   862
subclass unbounded_dense_linorder
haftmann@36301
   863
proof
haftmann@36301
   864
  fix x y :: 'a
haftmann@36301
   865
  from less_add_one show "\<exists>y. x < y" .. 
haftmann@36301
   866
  from less_add_one have "x + (- 1) < (x + 1) + (- 1)" by (rule add_strict_right_mono)
haftmann@54230
   867
  then have "x - 1 < x + 1 - 1" by simp
haftmann@36301
   868
  then have "x - 1 < x" by (simp add: algebra_simps)
haftmann@36301
   869
  then show "\<exists>y. y < x" ..
haftmann@36301
   870
  show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
haftmann@36301
   871
qed
haftmann@36301
   872
haftmann@36301
   873
lemma nonzero_abs_inverse:
haftmann@36301
   874
     "a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
haftmann@36301
   875
apply (auto simp add: neq_iff abs_if nonzero_inverse_minus_eq 
haftmann@36301
   876
                      negative_imp_inverse_negative)
haftmann@36301
   877
apply (blast intro: positive_imp_inverse_positive elim: less_asym) 
haftmann@36301
   878
done
haftmann@36301
   879
haftmann@36301
   880
lemma nonzero_abs_divide:
haftmann@36301
   881
     "b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
haftmann@36301
   882
  by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
haftmann@36301
   883
haftmann@36301
   884
lemma field_le_epsilon:
haftmann@36301
   885
  assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
haftmann@36301
   886
  shows "x \<le> y"
haftmann@36301
   887
proof (rule dense_le)
haftmann@36301
   888
  fix t assume "t < x"
haftmann@36301
   889
  hence "0 < x - t" by (simp add: less_diff_eq)
haftmann@36301
   890
  from e [OF this] have "x + 0 \<le> x + (y - t)" by (simp add: algebra_simps)
haftmann@36301
   891
  then have "0 \<le> y - t" by (simp only: add_le_cancel_left)
haftmann@36301
   892
  then show "t \<le> y" by (simp add: algebra_simps)
haftmann@36301
   893
qed
haftmann@36301
   894
haftmann@36301
   895
end
haftmann@36301
   896
haftmann@36414
   897
class linordered_field_inverse_zero = linordered_field + field_inverse_zero
haftmann@36348
   898
begin
haftmann@36348
   899
haftmann@36301
   900
lemma le_divide_eq:
haftmann@36301
   901
  "(a \<le> b/c) = 
haftmann@36301
   902
   (if 0 < c then a*c \<le> b
haftmann@36301
   903
             else if c < 0 then b \<le> a*c
haftmann@36409
   904
             else  a \<le> 0)"
haftmann@36301
   905
apply (cases "c=0", simp) 
haftmann@36301
   906
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
haftmann@36301
   907
done
haftmann@36301
   908
paulson@14277
   909
lemma inverse_positive_iff_positive [simp]:
haftmann@36409
   910
  "(0 < inverse a) = (0 < a)"
haftmann@21328
   911
apply (cases "a = 0", simp)
paulson@14277
   912
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
paulson@14277
   913
done
paulson@14277
   914
paulson@14277
   915
lemma inverse_negative_iff_negative [simp]:
haftmann@36409
   916
  "(inverse a < 0) = (a < 0)"
haftmann@21328
   917
apply (cases "a = 0", simp)
paulson@14277
   918
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
paulson@14277
   919
done
paulson@14277
   920
paulson@14277
   921
lemma inverse_nonnegative_iff_nonnegative [simp]:
haftmann@36409
   922
  "0 \<le> inverse a \<longleftrightarrow> 0 \<le> a"
haftmann@36409
   923
  by (simp add: not_less [symmetric])
paulson@14277
   924
paulson@14277
   925
lemma inverse_nonpositive_iff_nonpositive [simp]:
haftmann@36409
   926
  "inverse a \<le> 0 \<longleftrightarrow> a \<le> 0"
haftmann@36409
   927
  by (simp add: not_less [symmetric])
paulson@14277
   928
paulson@14365
   929
lemma one_less_inverse_iff:
haftmann@36409
   930
  "1 < inverse x \<longleftrightarrow> 0 < x \<and> x < 1"
nipkow@23482
   931
proof cases
paulson@14365
   932
  assume "0 < x"
paulson@14365
   933
    with inverse_less_iff_less [OF zero_less_one, of x]
paulson@14365
   934
    show ?thesis by simp
paulson@14365
   935
next
paulson@14365
   936
  assume notless: "~ (0 < x)"
paulson@14365
   937
  have "~ (1 < inverse x)"
paulson@14365
   938
  proof
wenzelm@53374
   939
    assume *: "1 < inverse x"
wenzelm@53374
   940
    also from notless and * have "... \<le> 0" by simp
paulson@14365
   941
    also have "... < 1" by (rule zero_less_one) 
paulson@14365
   942
    finally show False by auto
paulson@14365
   943
  qed
paulson@14365
   944
  with notless show ?thesis by simp
paulson@14365
   945
qed
paulson@14365
   946
paulson@14365
   947
lemma one_le_inverse_iff:
haftmann@36409
   948
  "1 \<le> inverse x \<longleftrightarrow> 0 < x \<and> x \<le> 1"
haftmann@36409
   949
proof (cases "x = 1")
haftmann@36409
   950
  case True then show ?thesis by simp
haftmann@36409
   951
next
haftmann@36409
   952
  case False then have "inverse x \<noteq> 1" by simp
haftmann@36409
   953
  then have "1 \<noteq> inverse x" by blast
haftmann@36409
   954
  then have "1 \<le> inverse x \<longleftrightarrow> 1 < inverse x" by (simp add: le_less)
haftmann@36409
   955
  with False show ?thesis by (auto simp add: one_less_inverse_iff)
haftmann@36409
   956
qed
paulson@14365
   957
paulson@14365
   958
lemma inverse_less_1_iff:
haftmann@36409
   959
  "inverse x < 1 \<longleftrightarrow> x \<le> 0 \<or> 1 < x"
haftmann@36409
   960
  by (simp add: not_le [symmetric] one_le_inverse_iff) 
paulson@14365
   961
paulson@14365
   962
lemma inverse_le_1_iff:
haftmann@36409
   963
  "inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x"
haftmann@36409
   964
  by (simp add: not_less [symmetric] one_less_inverse_iff) 
paulson@14365
   965
paulson@14288
   966
lemma divide_le_eq:
paulson@14288
   967
  "(b/c \<le> a) = 
paulson@14288
   968
   (if 0 < c then b \<le> a*c
paulson@14288
   969
             else if c < 0 then a*c \<le> b
haftmann@36409
   970
             else 0 \<le> a)"
haftmann@21328
   971
apply (cases "c=0", simp) 
haftmann@36409
   972
apply (force simp add: pos_divide_le_eq neg_divide_le_eq) 
paulson@14288
   973
done
paulson@14288
   974
paulson@14288
   975
lemma less_divide_eq:
paulson@14288
   976
  "(a < b/c) = 
paulson@14288
   977
   (if 0 < c then a*c < b
paulson@14288
   978
             else if c < 0 then b < a*c
haftmann@36409
   979
             else  a < 0)"
haftmann@21328
   980
apply (cases "c=0", simp) 
haftmann@36409
   981
apply (force simp add: pos_less_divide_eq neg_less_divide_eq) 
paulson@14288
   982
done
paulson@14288
   983
paulson@14288
   984
lemma divide_less_eq:
paulson@14288
   985
  "(b/c < a) = 
paulson@14288
   986
   (if 0 < c then b < a*c
paulson@14288
   987
             else if c < 0 then a*c < b
haftmann@36409
   988
             else 0 < a)"
haftmann@21328
   989
apply (cases "c=0", simp) 
haftmann@36409
   990
apply (force simp add: pos_divide_less_eq neg_divide_less_eq)
paulson@14288
   991
done
paulson@14288
   992
haftmann@36301
   993
text {*Division and Signs*}
avigad@16775
   994
avigad@16775
   995
lemma zero_less_divide_iff:
haftmann@36409
   996
     "(0 < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
avigad@16775
   997
by (simp add: divide_inverse zero_less_mult_iff)
avigad@16775
   998
avigad@16775
   999
lemma divide_less_0_iff:
haftmann@36409
  1000
     "(a/b < 0) = 
avigad@16775
  1001
      (0 < a & b < 0 | a < 0 & 0 < b)"
avigad@16775
  1002
by (simp add: divide_inverse mult_less_0_iff)
avigad@16775
  1003
avigad@16775
  1004
lemma zero_le_divide_iff:
haftmann@36409
  1005
     "(0 \<le> a/b) =
avigad@16775
  1006
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
avigad@16775
  1007
by (simp add: divide_inverse zero_le_mult_iff)
avigad@16775
  1008
avigad@16775
  1009
lemma divide_le_0_iff:
haftmann@36409
  1010
     "(a/b \<le> 0) =
avigad@16775
  1011
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
avigad@16775
  1012
by (simp add: divide_inverse mult_le_0_iff)
avigad@16775
  1013
haftmann@36301
  1014
text {* Division and the Number One *}
paulson@14353
  1015
paulson@14353
  1016
text{*Simplify expressions equated with 1*}
paulson@14353
  1017
blanchet@54147
  1018
lemma zero_eq_1_divide_iff [simp]:
haftmann@36409
  1019
     "(0 = 1/a) = (a = 0)"
nipkow@23482
  1020
apply (cases "a=0", simp)
nipkow@23482
  1021
apply (auto simp add: nonzero_eq_divide_eq)
paulson@14353
  1022
done
paulson@14353
  1023
blanchet@54147
  1024
lemma one_divide_eq_0_iff [simp]:
haftmann@36409
  1025
     "(1/a = 0) = (a = 0)"
nipkow@23482
  1026
apply (cases "a=0", simp)
nipkow@23482
  1027
apply (insert zero_neq_one [THEN not_sym])
nipkow@23482
  1028
apply (auto simp add: nonzero_divide_eq_eq)
paulson@14353
  1029
done
paulson@14353
  1030
paulson@14353
  1031
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
haftmann@36423
  1032
blanchet@54147
  1033
lemma zero_le_divide_1_iff [simp]:
haftmann@36423
  1034
  "0 \<le> 1 / a \<longleftrightarrow> 0 \<le> a"
haftmann@36423
  1035
  by (simp add: zero_le_divide_iff)
paulson@17085
  1036
blanchet@54147
  1037
lemma zero_less_divide_1_iff [simp]:
haftmann@36423
  1038
  "0 < 1 / a \<longleftrightarrow> 0 < a"
haftmann@36423
  1039
  by (simp add: zero_less_divide_iff)
haftmann@36423
  1040
blanchet@54147
  1041
lemma divide_le_0_1_iff [simp]:
haftmann@36423
  1042
  "1 / a \<le> 0 \<longleftrightarrow> a \<le> 0"
haftmann@36423
  1043
  by (simp add: divide_le_0_iff)
haftmann@36423
  1044
blanchet@54147
  1045
lemma divide_less_0_1_iff [simp]:
haftmann@36423
  1046
  "1 / a < 0 \<longleftrightarrow> a < 0"
haftmann@36423
  1047
  by (simp add: divide_less_0_iff)
paulson@14353
  1048
paulson@14293
  1049
lemma divide_right_mono:
haftmann@36409
  1050
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/c"
haftmann@36409
  1051
by (force simp add: divide_strict_right_mono le_less)
paulson@14293
  1052
haftmann@36409
  1053
lemma divide_right_mono_neg: "a <= b 
avigad@16775
  1054
    ==> c <= 0 ==> b / c <= a / c"
nipkow@23482
  1055
apply (drule divide_right_mono [of _ _ "- c"])
hoelzl@56479
  1056
apply auto
avigad@16775
  1057
done
avigad@16775
  1058
haftmann@36409
  1059
lemma divide_left_mono_neg: "a <= b 
avigad@16775
  1060
    ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
avigad@16775
  1061
  apply (drule divide_left_mono [of _ _ "- c"])
hoelzl@56479
  1062
  apply (auto simp add: mult_commute)
avigad@16775
  1063
done
avigad@16775
  1064
hoelzl@42904
  1065
lemma inverse_le_iff:
hoelzl@42904
  1066
  "inverse a \<le> inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b \<le> a) \<and> (a * b \<le> 0 \<longrightarrow> a \<le> b)"
hoelzl@42904
  1067
proof -
hoelzl@42904
  1068
  { assume "a < 0"
hoelzl@42904
  1069
    then have "inverse a < 0" by simp
hoelzl@42904
  1070
    moreover assume "0 < b"
hoelzl@42904
  1071
    then have "0 < inverse b" by simp
hoelzl@42904
  1072
    ultimately have "inverse a < inverse b" by (rule less_trans)
hoelzl@42904
  1073
    then have "inverse a \<le> inverse b" by simp }
hoelzl@42904
  1074
  moreover
hoelzl@42904
  1075
  { assume "b < 0"
hoelzl@42904
  1076
    then have "inverse b < 0" by simp
hoelzl@42904
  1077
    moreover assume "0 < a"
hoelzl@42904
  1078
    then have "0 < inverse a" by simp
hoelzl@42904
  1079
    ultimately have "inverse b < inverse a" by (rule less_trans)
hoelzl@42904
  1080
    then have "\<not> inverse a \<le> inverse b" by simp }
hoelzl@42904
  1081
  ultimately show ?thesis
hoelzl@42904
  1082
    by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
hoelzl@42904
  1083
       (auto simp: not_less zero_less_mult_iff mult_le_0_iff)
hoelzl@42904
  1084
qed
hoelzl@42904
  1085
hoelzl@42904
  1086
lemma inverse_less_iff:
hoelzl@42904
  1087
  "inverse a < inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b < a) \<and> (a * b \<le> 0 \<longrightarrow> a < b)"
hoelzl@42904
  1088
  by (subst less_le) (auto simp: inverse_le_iff)
hoelzl@42904
  1089
hoelzl@42904
  1090
lemma divide_le_cancel:
hoelzl@42904
  1091
  "a / c \<le> b / c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
hoelzl@42904
  1092
  by (simp add: divide_inverse mult_le_cancel_right)
hoelzl@42904
  1093
hoelzl@42904
  1094
lemma divide_less_cancel:
hoelzl@42904
  1095
  "a / c < b / c \<longleftrightarrow> (0 < c \<longrightarrow> a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0"
hoelzl@42904
  1096
  by (auto simp add: divide_inverse mult_less_cancel_right)
hoelzl@42904
  1097
avigad@16775
  1098
text{*Simplify quotients that are compared with the value 1.*}
avigad@16775
  1099
blanchet@54147
  1100
lemma le_divide_eq_1:
haftmann@36409
  1101
  "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
avigad@16775
  1102
by (auto simp add: le_divide_eq)
avigad@16775
  1103
blanchet@54147
  1104
lemma divide_le_eq_1:
haftmann@36409
  1105
  "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
avigad@16775
  1106
by (auto simp add: divide_le_eq)
avigad@16775
  1107
blanchet@54147
  1108
lemma less_divide_eq_1:
haftmann@36409
  1109
  "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
avigad@16775
  1110
by (auto simp add: less_divide_eq)
avigad@16775
  1111
blanchet@54147
  1112
lemma divide_less_eq_1:
haftmann@36409
  1113
  "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
avigad@16775
  1114
by (auto simp add: divide_less_eq)
avigad@16775
  1115
wenzelm@23389
  1116
haftmann@36301
  1117
text {*Conditional Simplification Rules: No Case Splits*}
avigad@16775
  1118
blanchet@54147
  1119
lemma le_divide_eq_1_pos [simp]:
haftmann@36409
  1120
  "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
avigad@16775
  1121
by (auto simp add: le_divide_eq)
avigad@16775
  1122
blanchet@54147
  1123
lemma le_divide_eq_1_neg [simp]:
haftmann@36409
  1124
  "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
avigad@16775
  1125
by (auto simp add: le_divide_eq)
avigad@16775
  1126
blanchet@54147
  1127
lemma divide_le_eq_1_pos [simp]:
haftmann@36409
  1128
  "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
avigad@16775
  1129
by (auto simp add: divide_le_eq)
avigad@16775
  1130
blanchet@54147
  1131
lemma divide_le_eq_1_neg [simp]:
haftmann@36409
  1132
  "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
avigad@16775
  1133
by (auto simp add: divide_le_eq)
avigad@16775
  1134
blanchet@54147
  1135
lemma less_divide_eq_1_pos [simp]:
haftmann@36409
  1136
  "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
avigad@16775
  1137
by (auto simp add: less_divide_eq)
avigad@16775
  1138
blanchet@54147
  1139
lemma less_divide_eq_1_neg [simp]:
haftmann@36409
  1140
  "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
avigad@16775
  1141
by (auto simp add: less_divide_eq)
avigad@16775
  1142
blanchet@54147
  1143
lemma divide_less_eq_1_pos [simp]:
haftmann@36409
  1144
  "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
paulson@18649
  1145
by (auto simp add: divide_less_eq)
paulson@18649
  1146
blanchet@54147
  1147
lemma divide_less_eq_1_neg [simp]:
haftmann@36409
  1148
  "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
avigad@16775
  1149
by (auto simp add: divide_less_eq)
avigad@16775
  1150
blanchet@54147
  1151
lemma eq_divide_eq_1 [simp]:
haftmann@36409
  1152
  "(1 = b/a) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1153
by (auto simp add: eq_divide_eq)
avigad@16775
  1154
blanchet@54147
  1155
lemma divide_eq_eq_1 [simp]:
haftmann@36409
  1156
  "(b/a = 1) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1157
by (auto simp add: divide_eq_eq)
avigad@16775
  1158
paulson@14294
  1159
lemma abs_inverse [simp]:
haftmann@36409
  1160
     "\<bar>inverse a\<bar> = 
haftmann@36301
  1161
      inverse \<bar>a\<bar>"
haftmann@21328
  1162
apply (cases "a=0", simp) 
paulson@14294
  1163
apply (simp add: nonzero_abs_inverse) 
paulson@14294
  1164
done
paulson@14294
  1165
paulson@15234
  1166
lemma abs_divide [simp]:
haftmann@36409
  1167
     "\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
haftmann@21328
  1168
apply (cases "b=0", simp) 
paulson@14294
  1169
apply (simp add: nonzero_abs_divide) 
paulson@14294
  1170
done
paulson@14294
  1171
haftmann@36409
  1172
lemma abs_div_pos: "0 < y ==> 
haftmann@36301
  1173
    \<bar>x\<bar> / y = \<bar>x / y\<bar>"
haftmann@25304
  1174
  apply (subst abs_divide)
haftmann@25304
  1175
  apply (simp add: order_less_imp_le)
haftmann@25304
  1176
done
avigad@16775
  1177
lp15@55718
  1178
lemma zero_le_divide_abs_iff [simp]: "(0 \<le> a / abs b) = (0 \<le> a | b = 0)" 
lp15@55718
  1179
by (auto simp: zero_le_divide_iff)
lp15@55718
  1180
lp15@55718
  1181
lemma divide_le_0_abs_iff [simp]: "(a / abs b \<le> 0) = (a \<le> 0 | b = 0)" 
lp15@55718
  1182
by (auto simp: divide_le_0_iff)
lp15@55718
  1183
hoelzl@35579
  1184
lemma field_le_mult_one_interval:
hoelzl@35579
  1185
  assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
hoelzl@35579
  1186
  shows "x \<le> y"
hoelzl@35579
  1187
proof (cases "0 < x")
hoelzl@35579
  1188
  assume "0 < x"
hoelzl@35579
  1189
  thus ?thesis
hoelzl@35579
  1190
    using dense_le_bounded[of 0 1 "y/x"] *
hoelzl@35579
  1191
    unfolding le_divide_eq if_P[OF `0 < x`] by simp
hoelzl@35579
  1192
next
hoelzl@35579
  1193
  assume "\<not>0 < x" hence "x \<le> 0" by simp
hoelzl@35579
  1194
  obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1\<Colon>'a"] by auto
hoelzl@35579
  1195
  hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] `x \<le> 0` by auto
hoelzl@35579
  1196
  also note *[OF s]
hoelzl@35579
  1197
  finally show ?thesis .
hoelzl@35579
  1198
qed
haftmann@35090
  1199
haftmann@36409
  1200
end
haftmann@36409
  1201
haftmann@52435
  1202
code_identifier
haftmann@52435
  1203
  code_module Fields \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  1204
paulson@14265
  1205
end