src/HOL/Fields.thy
author paulson <lp15@cam.ac.uk>
Thu Apr 03 23:51:52 2014 +0100 (2014-04-03)
changeset 56409 36489d77c484
parent 56365 713f9b9a7e51
child 56410 a14831ac3023
permissions -rw-r--r--
removing simprule status for divide_minus_left and divide_minus_right
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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: "(-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 add: divide_inverse)
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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]:
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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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using nonzero_mult_divide_mult_cancel_left [of 1 a b] by simp
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lemma nonzero_divide_mult_cancel_right [simp]:
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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]:
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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]:
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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]:
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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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  by (simp add: add_divide_distrib)
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   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
blanchet@54147
   386
lemma divide_divide_eq_right [simp]:
haftmann@36409
   387
  "a / (b / c) = (a * c) / b"
haftmann@36409
   388
  by (simp add: divide_inverse mult_ac)
paulson@14288
   389
blanchet@54147
   390
lemma divide_divide_eq_left [simp]:
haftmann@36409
   391
  "(a / b) / c = a / (b * c)"
haftmann@36409
   392
  by (simp add: divide_inverse mult_assoc)
paulson@14288
   393
lp15@56365
   394
lemma divide_divide_times_eq:
lp15@56365
   395
  "(x / y) / (z / w) = (x * w) / (y * z)"
lp15@56365
   396
  by simp
wenzelm@23389
   397
haftmann@36301
   398
text {*Special Cancellation Simprules for Division*}
paulson@15234
   399
blanchet@54147
   400
lemma mult_divide_mult_cancel_left_if [simp]:
haftmann@36409
   401
  shows "(c * a) / (c * b) = (if c = 0 then 0 else a / b)"
haftmann@36409
   402
  by (simp add: mult_divide_mult_cancel_left)
nipkow@23413
   403
paulson@15234
   404
haftmann@36301
   405
text {* Division and Unary Minus *}
paulson@14293
   406
haftmann@36409
   407
lemma minus_divide_right:
haftmann@36409
   408
  "- (a / b) = a / - b"
haftmann@36409
   409
  by (simp add: divide_inverse)
paulson@14430
   410
lp15@56409
   411
lemma divide_minus_right:
haftmann@36409
   412
  "a / - b = - (a / b)"
haftmann@36409
   413
  by (simp add: divide_inverse)
huffman@30630
   414
huffman@30630
   415
lemma minus_divide_divide:
haftmann@36409
   416
  "(- a) / (- b) = a / b"
haftmann@21328
   417
apply (cases "b=0", simp) 
paulson@14293
   418
apply (simp add: nonzero_minus_divide_divide) 
paulson@14293
   419
done
paulson@14293
   420
nipkow@23482
   421
lemma eq_divide_eq:
haftmann@36409
   422
  "a = b / c \<longleftrightarrow> (if c \<noteq> 0 then a * c = b else a = 0)"
haftmann@36409
   423
  by (simp add: nonzero_eq_divide_eq)
nipkow@23482
   424
nipkow@23482
   425
lemma divide_eq_eq:
haftmann@36409
   426
  "b / c = a \<longleftrightarrow> (if c \<noteq> 0 then b = a * c else a = 0)"
haftmann@36409
   427
  by (force simp add: nonzero_divide_eq_eq)
paulson@14293
   428
haftmann@36301
   429
lemma inverse_eq_1_iff [simp]:
haftmann@36409
   430
  "inverse x = 1 \<longleftrightarrow> x = 1"
haftmann@36409
   431
  by (insert inverse_eq_iff_eq [of x 1], simp) 
wenzelm@23389
   432
blanchet@54147
   433
lemma divide_eq_0_iff [simp]:
haftmann@36409
   434
  "a / b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
haftmann@36409
   435
  by (simp add: divide_inverse)
haftmann@36301
   436
blanchet@54147
   437
lemma divide_cancel_right [simp]:
haftmann@36409
   438
  "a / c = b / c \<longleftrightarrow> c = 0 \<or> a = b"
haftmann@36409
   439
  apply (cases "c=0", simp)
haftmann@36409
   440
  apply (simp add: divide_inverse)
haftmann@36409
   441
  done
haftmann@36301
   442
blanchet@54147
   443
lemma divide_cancel_left [simp]:
haftmann@36409
   444
  "c / a = c / b \<longleftrightarrow> c = 0 \<or> a = b" 
haftmann@36409
   445
  apply (cases "c=0", simp)
haftmann@36409
   446
  apply (simp add: divide_inverse)
haftmann@36409
   447
  done
haftmann@36301
   448
blanchet@54147
   449
lemma divide_eq_1_iff [simp]:
haftmann@36409
   450
  "a / b = 1 \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
haftmann@36409
   451
  apply (cases "b=0", simp)
haftmann@36409
   452
  apply (simp add: right_inverse_eq)
haftmann@36409
   453
  done
haftmann@36301
   454
blanchet@54147
   455
lemma one_eq_divide_iff [simp]:
haftmann@36409
   456
  "1 = a / b \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
haftmann@36409
   457
  by (simp add: eq_commute [of 1])
haftmann@36409
   458
haftmann@36719
   459
lemma times_divide_times_eq:
haftmann@36719
   460
  "(x / y) * (z / w) = (x * z) / (y * w)"
haftmann@36719
   461
  by simp
haftmann@36719
   462
haftmann@36719
   463
lemma add_frac_num:
haftmann@36719
   464
  "y \<noteq> 0 \<Longrightarrow> x / y + z = (x + z * y) / y"
haftmann@36719
   465
  by (simp add: add_divide_distrib)
haftmann@36719
   466
haftmann@36719
   467
lemma add_num_frac:
haftmann@36719
   468
  "y \<noteq> 0 \<Longrightarrow> z + x / y = (x + z * y) / y"
haftmann@36719
   469
  by (simp add: add_divide_distrib add.commute)
haftmann@36719
   470
haftmann@36409
   471
end
haftmann@36301
   472
haftmann@36301
   473
huffman@44064
   474
subsection {* Ordered fields *}
haftmann@36301
   475
haftmann@36301
   476
class linordered_field = field + linordered_idom
haftmann@36301
   477
begin
paulson@14268
   478
paulson@14277
   479
lemma positive_imp_inverse_positive: 
haftmann@36301
   480
  assumes a_gt_0: "0 < a" 
haftmann@36301
   481
  shows "0 < inverse a"
nipkow@23482
   482
proof -
paulson@14268
   483
  have "0 < a * inverse a" 
haftmann@36301
   484
    by (simp add: a_gt_0 [THEN less_imp_not_eq2])
paulson@14268
   485
  thus "0 < inverse a" 
haftmann@36301
   486
    by (simp add: a_gt_0 [THEN less_not_sym] zero_less_mult_iff)
nipkow@23482
   487
qed
paulson@14268
   488
paulson@14277
   489
lemma negative_imp_inverse_negative:
haftmann@36301
   490
  "a < 0 \<Longrightarrow> inverse a < 0"
haftmann@36301
   491
  by (insert positive_imp_inverse_positive [of "-a"], 
haftmann@36301
   492
    simp add: nonzero_inverse_minus_eq less_imp_not_eq)
paulson@14268
   493
paulson@14268
   494
lemma inverse_le_imp_le:
haftmann@36301
   495
  assumes invle: "inverse a \<le> inverse b" and apos: "0 < a"
haftmann@36301
   496
  shows "b \<le> a"
nipkow@23482
   497
proof (rule classical)
paulson@14268
   498
  assume "~ b \<le> a"
nipkow@23482
   499
  hence "a < b"  by (simp add: linorder_not_le)
haftmann@36301
   500
  hence bpos: "0 < b"  by (blast intro: apos less_trans)
paulson@14268
   501
  hence "a * inverse a \<le> a * inverse b"
haftmann@36301
   502
    by (simp add: apos invle less_imp_le mult_left_mono)
paulson@14268
   503
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
haftmann@36301
   504
    by (simp add: bpos less_imp_le mult_right_mono)
haftmann@36301
   505
  thus "b \<le> a"  by (simp add: mult_assoc apos bpos less_imp_not_eq2)
nipkow@23482
   506
qed
paulson@14268
   507
paulson@14277
   508
lemma inverse_positive_imp_positive:
haftmann@36301
   509
  assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
haftmann@36301
   510
  shows "0 < a"
wenzelm@23389
   511
proof -
paulson@14277
   512
  have "0 < inverse (inverse a)"
wenzelm@23389
   513
    using inv_gt_0 by (rule positive_imp_inverse_positive)
paulson@14277
   514
  thus "0 < a"
wenzelm@23389
   515
    using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
   516
qed
paulson@14277
   517
haftmann@36301
   518
lemma inverse_negative_imp_negative:
haftmann@36301
   519
  assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0"
haftmann@36301
   520
  shows "a < 0"
haftmann@36301
   521
proof -
haftmann@36301
   522
  have "inverse (inverse a) < 0"
haftmann@36301
   523
    using inv_less_0 by (rule negative_imp_inverse_negative)
haftmann@36301
   524
  thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
haftmann@36301
   525
qed
haftmann@36301
   526
haftmann@36301
   527
lemma linordered_field_no_lb:
haftmann@36301
   528
  "\<forall>x. \<exists>y. y < x"
haftmann@36301
   529
proof
haftmann@36301
   530
  fix x::'a
haftmann@36301
   531
  have m1: "- (1::'a) < 0" by simp
haftmann@36301
   532
  from add_strict_right_mono[OF m1, where c=x] 
haftmann@36301
   533
  have "(- 1) + x < x" by simp
haftmann@36301
   534
  thus "\<exists>y. y < x" by blast
haftmann@36301
   535
qed
haftmann@36301
   536
haftmann@36301
   537
lemma linordered_field_no_ub:
haftmann@36301
   538
  "\<forall> x. \<exists>y. y > x"
haftmann@36301
   539
proof
haftmann@36301
   540
  fix x::'a
haftmann@36301
   541
  have m1: " (1::'a) > 0" by simp
haftmann@36301
   542
  from add_strict_right_mono[OF m1, where c=x] 
haftmann@36301
   543
  have "1 + x > x" by simp
haftmann@36301
   544
  thus "\<exists>y. y > x" by blast
haftmann@36301
   545
qed
haftmann@36301
   546
haftmann@36301
   547
lemma less_imp_inverse_less:
haftmann@36301
   548
  assumes less: "a < b" and apos:  "0 < a"
haftmann@36301
   549
  shows "inverse b < inverse a"
haftmann@36301
   550
proof (rule ccontr)
haftmann@36301
   551
  assume "~ inverse b < inverse a"
haftmann@36301
   552
  hence "inverse a \<le> inverse b" by simp
haftmann@36301
   553
  hence "~ (a < b)"
haftmann@36301
   554
    by (simp add: not_less inverse_le_imp_le [OF _ apos])
haftmann@36301
   555
  thus False by (rule notE [OF _ less])
haftmann@36301
   556
qed
haftmann@36301
   557
haftmann@36301
   558
lemma inverse_less_imp_less:
haftmann@36301
   559
  "inverse a < inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b < a"
haftmann@36301
   560
apply (simp add: less_le [of "inverse a"] less_le [of "b"])
haftmann@36301
   561
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
haftmann@36301
   562
done
haftmann@36301
   563
haftmann@36301
   564
text{*Both premises are essential. Consider -1 and 1.*}
blanchet@54147
   565
lemma inverse_less_iff_less [simp]:
haftmann@36301
   566
  "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
haftmann@36301
   567
  by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
haftmann@36301
   568
haftmann@36301
   569
lemma le_imp_inverse_le:
haftmann@36301
   570
  "a \<le> b \<Longrightarrow> 0 < a \<Longrightarrow> inverse b \<le> inverse a"
haftmann@36301
   571
  by (force simp add: le_less less_imp_inverse_less)
haftmann@36301
   572
blanchet@54147
   573
lemma inverse_le_iff_le [simp]:
haftmann@36301
   574
  "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
haftmann@36301
   575
  by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
haftmann@36301
   576
haftmann@36301
   577
haftmann@36301
   578
text{*These results refer to both operands being negative.  The opposite-sign
haftmann@36301
   579
case is trivial, since inverse preserves signs.*}
haftmann@36301
   580
lemma inverse_le_imp_le_neg:
haftmann@36301
   581
  "inverse a \<le> inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a"
haftmann@36301
   582
apply (rule classical) 
haftmann@36301
   583
apply (subgoal_tac "a < 0") 
haftmann@36301
   584
 prefer 2 apply force
haftmann@36301
   585
apply (insert inverse_le_imp_le [of "-b" "-a"])
haftmann@36301
   586
apply (simp add: nonzero_inverse_minus_eq) 
haftmann@36301
   587
done
haftmann@36301
   588
haftmann@36301
   589
lemma less_imp_inverse_less_neg:
haftmann@36301
   590
   "a < b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b < inverse a"
haftmann@36301
   591
apply (subgoal_tac "a < 0") 
haftmann@36301
   592
 prefer 2 apply (blast intro: less_trans) 
haftmann@36301
   593
apply (insert less_imp_inverse_less [of "-b" "-a"])
haftmann@36301
   594
apply (simp add: nonzero_inverse_minus_eq) 
haftmann@36301
   595
done
haftmann@36301
   596
haftmann@36301
   597
lemma inverse_less_imp_less_neg:
haftmann@36301
   598
   "inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a"
haftmann@36301
   599
apply (rule classical) 
haftmann@36301
   600
apply (subgoal_tac "a < 0") 
haftmann@36301
   601
 prefer 2
haftmann@36301
   602
 apply force
haftmann@36301
   603
apply (insert inverse_less_imp_less [of "-b" "-a"])
haftmann@36301
   604
apply (simp add: nonzero_inverse_minus_eq) 
haftmann@36301
   605
done
haftmann@36301
   606
blanchet@54147
   607
lemma inverse_less_iff_less_neg [simp]:
haftmann@36301
   608
  "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
haftmann@36301
   609
apply (insert inverse_less_iff_less [of "-b" "-a"])
haftmann@36301
   610
apply (simp del: inverse_less_iff_less 
haftmann@36301
   611
            add: nonzero_inverse_minus_eq)
haftmann@36301
   612
done
haftmann@36301
   613
haftmann@36301
   614
lemma le_imp_inverse_le_neg:
haftmann@36301
   615
  "a \<le> b \<Longrightarrow> b < 0 ==> inverse b \<le> inverse a"
haftmann@36301
   616
  by (force simp add: le_less less_imp_inverse_less_neg)
haftmann@36301
   617
blanchet@54147
   618
lemma inverse_le_iff_le_neg [simp]:
haftmann@36301
   619
  "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
haftmann@36301
   620
  by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
haftmann@36301
   621
huffman@36774
   622
lemma one_less_inverse:
huffman@36774
   623
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> 1 < inverse a"
huffman@36774
   624
  using less_imp_inverse_less [of a 1, unfolded inverse_1] .
huffman@36774
   625
huffman@36774
   626
lemma one_le_inverse:
huffman@36774
   627
  "0 < a \<Longrightarrow> a \<le> 1 \<Longrightarrow> 1 \<le> inverse a"
huffman@36774
   628
  using le_imp_inverse_le [of a 1, unfolded inverse_1] .
huffman@36774
   629
haftmann@36348
   630
lemma pos_le_divide_eq [field_simps]: "0 < c ==> (a \<le> b/c) = (a*c \<le> b)"
haftmann@36301
   631
proof -
haftmann@36301
   632
  assume less: "0<c"
haftmann@36301
   633
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
haftmann@36304
   634
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   635
  also have "... = (a*c \<le> b)"
haftmann@36301
   636
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
haftmann@36301
   637
  finally show ?thesis .
haftmann@36301
   638
qed
haftmann@36301
   639
haftmann@36348
   640
lemma neg_le_divide_eq [field_simps]: "c < 0 ==> (a \<le> b/c) = (b \<le> a*c)"
haftmann@36301
   641
proof -
haftmann@36301
   642
  assume less: "c<0"
haftmann@36301
   643
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
haftmann@36304
   644
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   645
  also have "... = (b \<le> a*c)"
haftmann@36301
   646
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
haftmann@36301
   647
  finally show ?thesis .
haftmann@36301
   648
qed
haftmann@36301
   649
haftmann@36348
   650
lemma pos_less_divide_eq [field_simps]:
haftmann@36301
   651
     "0 < c ==> (a < b/c) = (a*c < b)"
haftmann@36301
   652
proof -
haftmann@36301
   653
  assume less: "0<c"
haftmann@36301
   654
  hence "(a < b/c) = (a*c < (b/c)*c)"
haftmann@36304
   655
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   656
  also have "... = (a*c < b)"
haftmann@36301
   657
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
haftmann@36301
   658
  finally show ?thesis .
haftmann@36301
   659
qed
haftmann@36301
   660
haftmann@36348
   661
lemma neg_less_divide_eq [field_simps]:
haftmann@36301
   662
 "c < 0 ==> (a < b/c) = (b < a*c)"
haftmann@36301
   663
proof -
haftmann@36301
   664
  assume less: "c<0"
haftmann@36301
   665
  hence "(a < b/c) = ((b/c)*c < a*c)"
haftmann@36304
   666
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   667
  also have "... = (b < a*c)"
haftmann@36301
   668
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
haftmann@36301
   669
  finally show ?thesis .
haftmann@36301
   670
qed
haftmann@36301
   671
haftmann@36348
   672
lemma pos_divide_less_eq [field_simps]:
haftmann@36301
   673
     "0 < c ==> (b/c < a) = (b < a*c)"
haftmann@36301
   674
proof -
haftmann@36301
   675
  assume less: "0<c"
haftmann@36301
   676
  hence "(b/c < a) = ((b/c)*c < a*c)"
haftmann@36304
   677
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   678
  also have "... = (b < a*c)"
haftmann@36301
   679
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
haftmann@36301
   680
  finally show ?thesis .
haftmann@36301
   681
qed
haftmann@36301
   682
haftmann@36348
   683
lemma neg_divide_less_eq [field_simps]:
haftmann@36301
   684
 "c < 0 ==> (b/c < a) = (a*c < b)"
haftmann@36301
   685
proof -
haftmann@36301
   686
  assume less: "c<0"
haftmann@36301
   687
  hence "(b/c < a) = (a*c < (b/c)*c)"
haftmann@36304
   688
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   689
  also have "... = (a*c < b)"
haftmann@36301
   690
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
haftmann@36301
   691
  finally show ?thesis .
haftmann@36301
   692
qed
haftmann@36301
   693
haftmann@36348
   694
lemma pos_divide_le_eq [field_simps]: "0 < c ==> (b/c \<le> a) = (b \<le> a*c)"
haftmann@36301
   695
proof -
haftmann@36301
   696
  assume less: "0<c"
haftmann@36301
   697
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
haftmann@36304
   698
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   699
  also have "... = (b \<le> a*c)"
haftmann@36301
   700
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
haftmann@36301
   701
  finally show ?thesis .
haftmann@36301
   702
qed
haftmann@36301
   703
haftmann@36348
   704
lemma neg_divide_le_eq [field_simps]: "c < 0 ==> (b/c \<le> a) = (a*c \<le> b)"
haftmann@36301
   705
proof -
haftmann@36301
   706
  assume less: "c<0"
haftmann@36301
   707
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
haftmann@36304
   708
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
haftmann@36301
   709
  also have "... = (a*c \<le> b)"
haftmann@36301
   710
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
haftmann@36301
   711
  finally show ?thesis .
haftmann@36301
   712
qed
haftmann@36301
   713
lp15@56365
   714
lemma frac_less_eq:
lp15@56365
   715
  "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y < w / z \<longleftrightarrow> (x * z - w * y) / (y * z) < 0"
lp15@56365
   716
  by (subst less_iff_diff_less_0) (simp add: diff_frac_eq )
lp15@56365
   717
lp15@56365
   718
lemma frac_le_eq:
lp15@56365
   719
  "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y \<le> w / z \<longleftrightarrow> (x * z - w * y) / (y * z) \<le> 0"
lp15@56365
   720
  by (subst le_iff_diff_le_0) (simp add: diff_frac_eq )
lp15@56365
   721
haftmann@36301
   722
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
haftmann@36301
   723
of positivity/negativity needed for @{text field_simps}. Have not added @{text
haftmann@36301
   724
sign_simps} to @{text field_simps} because the former can lead to case
haftmann@36301
   725
explosions. *}
haftmann@36301
   726
blanchet@54147
   727
lemmas sign_simps = algebra_simps zero_less_mult_iff mult_less_0_iff
haftmann@36348
   728
blanchet@54147
   729
lemmas (in -) sign_simps = algebra_simps zero_less_mult_iff mult_less_0_iff
haftmann@36301
   730
haftmann@36301
   731
(* Only works once linear arithmetic is installed:
haftmann@36301
   732
text{*An example:*}
haftmann@36301
   733
lemma fixes a b c d e f :: "'a::linordered_field"
haftmann@36301
   734
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
haftmann@36301
   735
 ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
haftmann@36301
   736
 ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
haftmann@36301
   737
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
haftmann@36301
   738
 prefer 2 apply(simp add:sign_simps)
haftmann@36301
   739
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
haftmann@36301
   740
 prefer 2 apply(simp add:sign_simps)
haftmann@36301
   741
apply(simp add:field_simps)
haftmann@36301
   742
done
haftmann@36301
   743
*)
haftmann@36301
   744
haftmann@36301
   745
lemma divide_pos_pos:
haftmann@36301
   746
  "0 < x ==> 0 < y ==> 0 < x / y"
haftmann@36301
   747
by(simp add:field_simps)
haftmann@36301
   748
haftmann@36301
   749
lemma divide_nonneg_pos:
haftmann@36301
   750
  "0 <= x ==> 0 < y ==> 0 <= x / y"
haftmann@36301
   751
by(simp add:field_simps)
haftmann@36301
   752
haftmann@36301
   753
lemma divide_neg_pos:
haftmann@36301
   754
  "x < 0 ==> 0 < y ==> x / y < 0"
haftmann@36301
   755
by(simp add:field_simps)
haftmann@36301
   756
haftmann@36301
   757
lemma divide_nonpos_pos:
haftmann@36301
   758
  "x <= 0 ==> 0 < y ==> x / y <= 0"
haftmann@36301
   759
by(simp add:field_simps)
haftmann@36301
   760
haftmann@36301
   761
lemma divide_pos_neg:
haftmann@36301
   762
  "0 < x ==> y < 0 ==> x / y < 0"
haftmann@36301
   763
by(simp add:field_simps)
haftmann@36301
   764
haftmann@36301
   765
lemma divide_nonneg_neg:
haftmann@36301
   766
  "0 <= x ==> y < 0 ==> x / y <= 0" 
haftmann@36301
   767
by(simp add:field_simps)
haftmann@36301
   768
haftmann@36301
   769
lemma divide_neg_neg:
haftmann@36301
   770
  "x < 0 ==> y < 0 ==> 0 < x / y"
haftmann@36301
   771
by(simp add:field_simps)
haftmann@36301
   772
haftmann@36301
   773
lemma divide_nonpos_neg:
haftmann@36301
   774
  "x <= 0 ==> y < 0 ==> 0 <= x / y"
haftmann@36301
   775
by(simp add:field_simps)
haftmann@36301
   776
haftmann@36301
   777
lemma divide_strict_right_mono:
haftmann@36301
   778
     "[|a < b; 0 < c|] ==> a / c < b / c"
haftmann@36301
   779
by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono 
haftmann@36301
   780
              positive_imp_inverse_positive)
haftmann@36301
   781
haftmann@36301
   782
haftmann@36301
   783
lemma divide_strict_right_mono_neg:
haftmann@36301
   784
     "[|b < a; c < 0|] ==> a / c < b / c"
haftmann@36301
   785
apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
haftmann@36301
   786
apply (simp add: less_imp_not_eq nonzero_minus_divide_right [symmetric])
haftmann@36301
   787
done
haftmann@36301
   788
haftmann@36301
   789
text{*The last premise ensures that @{term a} and @{term b} 
haftmann@36301
   790
      have the same sign*}
haftmann@36301
   791
lemma divide_strict_left_mono:
haftmann@36301
   792
  "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / b"
huffman@44921
   793
  by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono)
haftmann@36301
   794
haftmann@36301
   795
lemma divide_left_mono:
haftmann@36301
   796
  "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / b"
huffman@44921
   797
  by (auto simp: field_simps zero_less_mult_iff mult_right_mono)
haftmann@36301
   798
haftmann@36301
   799
lemma divide_strict_left_mono_neg:
haftmann@36301
   800
  "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / b"
huffman@44921
   801
  by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono_neg)
haftmann@36301
   802
haftmann@36301
   803
lemma mult_imp_div_pos_le: "0 < y ==> x <= z * y ==>
haftmann@36301
   804
    x / y <= z"
haftmann@36301
   805
by (subst pos_divide_le_eq, assumption+)
haftmann@36301
   806
haftmann@36301
   807
lemma mult_imp_le_div_pos: "0 < y ==> z * y <= x ==>
haftmann@36301
   808
    z <= x / y"
haftmann@36301
   809
by(simp add:field_simps)
haftmann@36301
   810
haftmann@36301
   811
lemma mult_imp_div_pos_less: "0 < y ==> x < z * y ==>
haftmann@36301
   812
    x / y < z"
haftmann@36301
   813
by(simp add:field_simps)
haftmann@36301
   814
haftmann@36301
   815
lemma mult_imp_less_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 frac_le: "0 <= x ==> 
haftmann@36301
   820
    x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
haftmann@36301
   821
  apply (rule mult_imp_div_pos_le)
haftmann@36301
   822
  apply simp
haftmann@36301
   823
  apply (subst times_divide_eq_left)
haftmann@36301
   824
  apply (rule mult_imp_le_div_pos, assumption)
haftmann@36301
   825
  apply (rule mult_mono)
haftmann@36301
   826
  apply simp_all
haftmann@36301
   827
done
haftmann@36301
   828
haftmann@36301
   829
lemma frac_less: "0 <= x ==> 
haftmann@36301
   830
    x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
haftmann@36301
   831
  apply (rule mult_imp_div_pos_less)
haftmann@36301
   832
  apply simp
haftmann@36301
   833
  apply (subst times_divide_eq_left)
haftmann@36301
   834
  apply (rule mult_imp_less_div_pos, assumption)
haftmann@36301
   835
  apply (erule mult_less_le_imp_less)
haftmann@36301
   836
  apply simp_all
haftmann@36301
   837
done
haftmann@36301
   838
haftmann@36301
   839
lemma frac_less2: "0 < x ==> 
haftmann@36301
   840
    x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
haftmann@36301
   841
  apply (rule mult_imp_div_pos_less)
haftmann@36301
   842
  apply simp_all
haftmann@36301
   843
  apply (rule mult_imp_less_div_pos, assumption)
haftmann@36301
   844
  apply (erule mult_le_less_imp_less)
haftmann@36301
   845
  apply simp_all
haftmann@36301
   846
done
haftmann@36301
   847
haftmann@36301
   848
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1)"
haftmann@36301
   849
by (simp add: field_simps zero_less_two)
haftmann@36301
   850
haftmann@36301
   851
lemma gt_half_sum: "a < b ==> (a+b)/(1+1) < b"
haftmann@36301
   852
by (simp add: field_simps zero_less_two)
haftmann@36301
   853
hoelzl@53215
   854
subclass unbounded_dense_linorder
haftmann@36301
   855
proof
haftmann@36301
   856
  fix x y :: 'a
haftmann@36301
   857
  from less_add_one show "\<exists>y. x < y" .. 
haftmann@36301
   858
  from less_add_one have "x + (- 1) < (x + 1) + (- 1)" by (rule add_strict_right_mono)
haftmann@54230
   859
  then have "x - 1 < x + 1 - 1" by simp
haftmann@36301
   860
  then have "x - 1 < x" by (simp add: algebra_simps)
haftmann@36301
   861
  then show "\<exists>y. y < x" ..
haftmann@36301
   862
  show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
haftmann@36301
   863
qed
haftmann@36301
   864
haftmann@36301
   865
lemma nonzero_abs_inverse:
haftmann@36301
   866
     "a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
haftmann@36301
   867
apply (auto simp add: neq_iff abs_if nonzero_inverse_minus_eq 
haftmann@36301
   868
                      negative_imp_inverse_negative)
haftmann@36301
   869
apply (blast intro: positive_imp_inverse_positive elim: less_asym) 
haftmann@36301
   870
done
haftmann@36301
   871
haftmann@36301
   872
lemma nonzero_abs_divide:
haftmann@36301
   873
     "b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
haftmann@36301
   874
  by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
haftmann@36301
   875
haftmann@36301
   876
lemma field_le_epsilon:
haftmann@36301
   877
  assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
haftmann@36301
   878
  shows "x \<le> y"
haftmann@36301
   879
proof (rule dense_le)
haftmann@36301
   880
  fix t assume "t < x"
haftmann@36301
   881
  hence "0 < x - t" by (simp add: less_diff_eq)
haftmann@36301
   882
  from e [OF this] have "x + 0 \<le> x + (y - t)" by (simp add: algebra_simps)
haftmann@36301
   883
  then have "0 \<le> y - t" by (simp only: add_le_cancel_left)
haftmann@36301
   884
  then show "t \<le> y" by (simp add: algebra_simps)
haftmann@36301
   885
qed
haftmann@36301
   886
haftmann@36301
   887
end
haftmann@36301
   888
haftmann@36414
   889
class linordered_field_inverse_zero = linordered_field + field_inverse_zero
haftmann@36348
   890
begin
haftmann@36348
   891
haftmann@36301
   892
lemma le_divide_eq:
haftmann@36301
   893
  "(a \<le> b/c) = 
haftmann@36301
   894
   (if 0 < c then a*c \<le> b
haftmann@36301
   895
             else if c < 0 then b \<le> a*c
haftmann@36409
   896
             else  a \<le> 0)"
haftmann@36301
   897
apply (cases "c=0", simp) 
haftmann@36301
   898
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
haftmann@36301
   899
done
haftmann@36301
   900
paulson@14277
   901
lemma inverse_positive_iff_positive [simp]:
haftmann@36409
   902
  "(0 < inverse a) = (0 < a)"
haftmann@21328
   903
apply (cases "a = 0", simp)
paulson@14277
   904
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
paulson@14277
   905
done
paulson@14277
   906
paulson@14277
   907
lemma inverse_negative_iff_negative [simp]:
haftmann@36409
   908
  "(inverse a < 0) = (a < 0)"
haftmann@21328
   909
apply (cases "a = 0", simp)
paulson@14277
   910
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
paulson@14277
   911
done
paulson@14277
   912
paulson@14277
   913
lemma inverse_nonnegative_iff_nonnegative [simp]:
haftmann@36409
   914
  "0 \<le> inverse a \<longleftrightarrow> 0 \<le> a"
haftmann@36409
   915
  by (simp add: not_less [symmetric])
paulson@14277
   916
paulson@14277
   917
lemma inverse_nonpositive_iff_nonpositive [simp]:
haftmann@36409
   918
  "inverse a \<le> 0 \<longleftrightarrow> a \<le> 0"
haftmann@36409
   919
  by (simp add: not_less [symmetric])
paulson@14277
   920
paulson@14365
   921
lemma one_less_inverse_iff:
haftmann@36409
   922
  "1 < inverse x \<longleftrightarrow> 0 < x \<and> x < 1"
nipkow@23482
   923
proof cases
paulson@14365
   924
  assume "0 < x"
paulson@14365
   925
    with inverse_less_iff_less [OF zero_less_one, of x]
paulson@14365
   926
    show ?thesis by simp
paulson@14365
   927
next
paulson@14365
   928
  assume notless: "~ (0 < x)"
paulson@14365
   929
  have "~ (1 < inverse x)"
paulson@14365
   930
  proof
wenzelm@53374
   931
    assume *: "1 < inverse x"
wenzelm@53374
   932
    also from notless and * have "... \<le> 0" by simp
paulson@14365
   933
    also have "... < 1" by (rule zero_less_one) 
paulson@14365
   934
    finally show False by auto
paulson@14365
   935
  qed
paulson@14365
   936
  with notless show ?thesis by simp
paulson@14365
   937
qed
paulson@14365
   938
paulson@14365
   939
lemma one_le_inverse_iff:
haftmann@36409
   940
  "1 \<le> inverse x \<longleftrightarrow> 0 < x \<and> x \<le> 1"
haftmann@36409
   941
proof (cases "x = 1")
haftmann@36409
   942
  case True then show ?thesis by simp
haftmann@36409
   943
next
haftmann@36409
   944
  case False then have "inverse x \<noteq> 1" by simp
haftmann@36409
   945
  then have "1 \<noteq> inverse x" by blast
haftmann@36409
   946
  then have "1 \<le> inverse x \<longleftrightarrow> 1 < inverse x" by (simp add: le_less)
haftmann@36409
   947
  with False show ?thesis by (auto simp add: one_less_inverse_iff)
haftmann@36409
   948
qed
paulson@14365
   949
paulson@14365
   950
lemma inverse_less_1_iff:
haftmann@36409
   951
  "inverse x < 1 \<longleftrightarrow> x \<le> 0 \<or> 1 < x"
haftmann@36409
   952
  by (simp add: not_le [symmetric] one_le_inverse_iff) 
paulson@14365
   953
paulson@14365
   954
lemma inverse_le_1_iff:
haftmann@36409
   955
  "inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x"
haftmann@36409
   956
  by (simp add: not_less [symmetric] one_less_inverse_iff) 
paulson@14365
   957
paulson@14288
   958
lemma divide_le_eq:
paulson@14288
   959
  "(b/c \<le> a) = 
paulson@14288
   960
   (if 0 < c then b \<le> a*c
paulson@14288
   961
             else if c < 0 then a*c \<le> b
haftmann@36409
   962
             else 0 \<le> a)"
haftmann@21328
   963
apply (cases "c=0", simp) 
haftmann@36409
   964
apply (force simp add: pos_divide_le_eq neg_divide_le_eq) 
paulson@14288
   965
done
paulson@14288
   966
paulson@14288
   967
lemma less_divide_eq:
paulson@14288
   968
  "(a < b/c) = 
paulson@14288
   969
   (if 0 < c then a*c < b
paulson@14288
   970
             else if c < 0 then b < a*c
haftmann@36409
   971
             else  a < 0)"
haftmann@21328
   972
apply (cases "c=0", simp) 
haftmann@36409
   973
apply (force simp add: pos_less_divide_eq neg_less_divide_eq) 
paulson@14288
   974
done
paulson@14288
   975
paulson@14288
   976
lemma divide_less_eq:
paulson@14288
   977
  "(b/c < a) = 
paulson@14288
   978
   (if 0 < c then b < a*c
paulson@14288
   979
             else if c < 0 then a*c < b
haftmann@36409
   980
             else 0 < a)"
haftmann@21328
   981
apply (cases "c=0", simp) 
haftmann@36409
   982
apply (force simp add: pos_divide_less_eq neg_divide_less_eq)
paulson@14288
   983
done
paulson@14288
   984
haftmann@36301
   985
text {*Division and Signs*}
avigad@16775
   986
avigad@16775
   987
lemma zero_less_divide_iff:
haftmann@36409
   988
     "(0 < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
avigad@16775
   989
by (simp add: divide_inverse zero_less_mult_iff)
avigad@16775
   990
avigad@16775
   991
lemma divide_less_0_iff:
haftmann@36409
   992
     "(a/b < 0) = 
avigad@16775
   993
      (0 < a & b < 0 | a < 0 & 0 < b)"
avigad@16775
   994
by (simp add: divide_inverse mult_less_0_iff)
avigad@16775
   995
avigad@16775
   996
lemma zero_le_divide_iff:
haftmann@36409
   997
     "(0 \<le> a/b) =
avigad@16775
   998
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
avigad@16775
   999
by (simp add: divide_inverse zero_le_mult_iff)
avigad@16775
  1000
avigad@16775
  1001
lemma divide_le_0_iff:
haftmann@36409
  1002
     "(a/b \<le> 0) =
avigad@16775
  1003
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
avigad@16775
  1004
by (simp add: divide_inverse mult_le_0_iff)
avigad@16775
  1005
haftmann@36301
  1006
text {* Division and the Number One *}
paulson@14353
  1007
paulson@14353
  1008
text{*Simplify expressions equated with 1*}
paulson@14353
  1009
blanchet@54147
  1010
lemma zero_eq_1_divide_iff [simp]:
haftmann@36409
  1011
     "(0 = 1/a) = (a = 0)"
nipkow@23482
  1012
apply (cases "a=0", simp)
nipkow@23482
  1013
apply (auto simp add: nonzero_eq_divide_eq)
paulson@14353
  1014
done
paulson@14353
  1015
blanchet@54147
  1016
lemma one_divide_eq_0_iff [simp]:
haftmann@36409
  1017
     "(1/a = 0) = (a = 0)"
nipkow@23482
  1018
apply (cases "a=0", simp)
nipkow@23482
  1019
apply (insert zero_neq_one [THEN not_sym])
nipkow@23482
  1020
apply (auto simp add: nonzero_divide_eq_eq)
paulson@14353
  1021
done
paulson@14353
  1022
paulson@14353
  1023
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
haftmann@36423
  1024
blanchet@54147
  1025
lemma zero_le_divide_1_iff [simp]:
haftmann@36423
  1026
  "0 \<le> 1 / a \<longleftrightarrow> 0 \<le> a"
haftmann@36423
  1027
  by (simp add: zero_le_divide_iff)
paulson@17085
  1028
blanchet@54147
  1029
lemma zero_less_divide_1_iff [simp]:
haftmann@36423
  1030
  "0 < 1 / a \<longleftrightarrow> 0 < a"
haftmann@36423
  1031
  by (simp add: zero_less_divide_iff)
haftmann@36423
  1032
blanchet@54147
  1033
lemma divide_le_0_1_iff [simp]:
haftmann@36423
  1034
  "1 / a \<le> 0 \<longleftrightarrow> a \<le> 0"
haftmann@36423
  1035
  by (simp add: divide_le_0_iff)
haftmann@36423
  1036
blanchet@54147
  1037
lemma divide_less_0_1_iff [simp]:
haftmann@36423
  1038
  "1 / a < 0 \<longleftrightarrow> a < 0"
haftmann@36423
  1039
  by (simp add: divide_less_0_iff)
paulson@14353
  1040
paulson@14293
  1041
lemma divide_right_mono:
haftmann@36409
  1042
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/c"
haftmann@36409
  1043
by (force simp add: divide_strict_right_mono le_less)
paulson@14293
  1044
haftmann@36409
  1045
lemma divide_right_mono_neg: "a <= b 
avigad@16775
  1046
    ==> c <= 0 ==> b / c <= a / c"
nipkow@23482
  1047
apply (drule divide_right_mono [of _ _ "- c"])
lp15@56409
  1048
apply (auto simp: divide_minus_right)
avigad@16775
  1049
done
avigad@16775
  1050
haftmann@36409
  1051
lemma divide_left_mono_neg: "a <= b 
avigad@16775
  1052
    ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
avigad@16775
  1053
  apply (drule divide_left_mono [of _ _ "- c"])
lp15@56409
  1054
  apply (auto simp add: divide_minus_left mult_commute)
avigad@16775
  1055
done
avigad@16775
  1056
hoelzl@42904
  1057
lemma inverse_le_iff:
hoelzl@42904
  1058
  "inverse a \<le> inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b \<le> a) \<and> (a * b \<le> 0 \<longrightarrow> a \<le> b)"
hoelzl@42904
  1059
proof -
hoelzl@42904
  1060
  { assume "a < 0"
hoelzl@42904
  1061
    then have "inverse a < 0" by simp
hoelzl@42904
  1062
    moreover assume "0 < b"
hoelzl@42904
  1063
    then have "0 < inverse b" by simp
hoelzl@42904
  1064
    ultimately have "inverse a < inverse b" by (rule less_trans)
hoelzl@42904
  1065
    then have "inverse a \<le> inverse b" by simp }
hoelzl@42904
  1066
  moreover
hoelzl@42904
  1067
  { assume "b < 0"
hoelzl@42904
  1068
    then have "inverse b < 0" by simp
hoelzl@42904
  1069
    moreover assume "0 < a"
hoelzl@42904
  1070
    then have "0 < inverse a" by simp
hoelzl@42904
  1071
    ultimately have "inverse b < inverse a" by (rule less_trans)
hoelzl@42904
  1072
    then have "\<not> inverse a \<le> inverse b" by simp }
hoelzl@42904
  1073
  ultimately show ?thesis
hoelzl@42904
  1074
    by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
hoelzl@42904
  1075
       (auto simp: not_less zero_less_mult_iff mult_le_0_iff)
hoelzl@42904
  1076
qed
hoelzl@42904
  1077
hoelzl@42904
  1078
lemma inverse_less_iff:
hoelzl@42904
  1079
  "inverse a < inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b < a) \<and> (a * b \<le> 0 \<longrightarrow> a < b)"
hoelzl@42904
  1080
  by (subst less_le) (auto simp: inverse_le_iff)
hoelzl@42904
  1081
hoelzl@42904
  1082
lemma divide_le_cancel:
hoelzl@42904
  1083
  "a / c \<le> b / c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
hoelzl@42904
  1084
  by (simp add: divide_inverse mult_le_cancel_right)
hoelzl@42904
  1085
hoelzl@42904
  1086
lemma divide_less_cancel:
hoelzl@42904
  1087
  "a / c < b / c \<longleftrightarrow> (0 < c \<longrightarrow> a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0"
hoelzl@42904
  1088
  by (auto simp add: divide_inverse mult_less_cancel_right)
hoelzl@42904
  1089
avigad@16775
  1090
text{*Simplify quotients that are compared with the value 1.*}
avigad@16775
  1091
blanchet@54147
  1092
lemma le_divide_eq_1:
haftmann@36409
  1093
  "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
avigad@16775
  1094
by (auto simp add: le_divide_eq)
avigad@16775
  1095
blanchet@54147
  1096
lemma divide_le_eq_1:
haftmann@36409
  1097
  "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
avigad@16775
  1098
by (auto simp add: divide_le_eq)
avigad@16775
  1099
blanchet@54147
  1100
lemma less_divide_eq_1:
haftmann@36409
  1101
  "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
avigad@16775
  1102
by (auto simp add: less_divide_eq)
avigad@16775
  1103
blanchet@54147
  1104
lemma divide_less_eq_1:
haftmann@36409
  1105
  "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
avigad@16775
  1106
by (auto simp add: divide_less_eq)
avigad@16775
  1107
wenzelm@23389
  1108
haftmann@36301
  1109
text {*Conditional Simplification Rules: No Case Splits*}
avigad@16775
  1110
blanchet@54147
  1111
lemma le_divide_eq_1_pos [simp]:
haftmann@36409
  1112
  "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
avigad@16775
  1113
by (auto simp add: le_divide_eq)
avigad@16775
  1114
blanchet@54147
  1115
lemma le_divide_eq_1_neg [simp]:
haftmann@36409
  1116
  "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
avigad@16775
  1117
by (auto simp add: le_divide_eq)
avigad@16775
  1118
blanchet@54147
  1119
lemma divide_le_eq_1_pos [simp]:
haftmann@36409
  1120
  "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
avigad@16775
  1121
by (auto simp add: divide_le_eq)
avigad@16775
  1122
blanchet@54147
  1123
lemma divide_le_eq_1_neg [simp]:
haftmann@36409
  1124
  "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
avigad@16775
  1125
by (auto simp add: divide_le_eq)
avigad@16775
  1126
blanchet@54147
  1127
lemma less_divide_eq_1_pos [simp]:
haftmann@36409
  1128
  "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
avigad@16775
  1129
by (auto simp add: less_divide_eq)
avigad@16775
  1130
blanchet@54147
  1131
lemma less_divide_eq_1_neg [simp]:
haftmann@36409
  1132
  "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
avigad@16775
  1133
by (auto simp add: less_divide_eq)
avigad@16775
  1134
blanchet@54147
  1135
lemma divide_less_eq_1_pos [simp]:
haftmann@36409
  1136
  "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
paulson@18649
  1137
by (auto simp add: divide_less_eq)
paulson@18649
  1138
blanchet@54147
  1139
lemma divide_less_eq_1_neg [simp]:
haftmann@36409
  1140
  "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
avigad@16775
  1141
by (auto simp add: divide_less_eq)
avigad@16775
  1142
blanchet@54147
  1143
lemma eq_divide_eq_1 [simp]:
haftmann@36409
  1144
  "(1 = b/a) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1145
by (auto simp add: eq_divide_eq)
avigad@16775
  1146
blanchet@54147
  1147
lemma divide_eq_eq_1 [simp]:
haftmann@36409
  1148
  "(b/a = 1) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1149
by (auto simp add: divide_eq_eq)
avigad@16775
  1150
paulson@14294
  1151
lemma abs_inverse [simp]:
haftmann@36409
  1152
     "\<bar>inverse a\<bar> = 
haftmann@36301
  1153
      inverse \<bar>a\<bar>"
haftmann@21328
  1154
apply (cases "a=0", simp) 
paulson@14294
  1155
apply (simp add: nonzero_abs_inverse) 
paulson@14294
  1156
done
paulson@14294
  1157
paulson@15234
  1158
lemma abs_divide [simp]:
haftmann@36409
  1159
     "\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
haftmann@21328
  1160
apply (cases "b=0", simp) 
paulson@14294
  1161
apply (simp add: nonzero_abs_divide) 
paulson@14294
  1162
done
paulson@14294
  1163
haftmann@36409
  1164
lemma abs_div_pos: "0 < y ==> 
haftmann@36301
  1165
    \<bar>x\<bar> / y = \<bar>x / y\<bar>"
haftmann@25304
  1166
  apply (subst abs_divide)
haftmann@25304
  1167
  apply (simp add: order_less_imp_le)
haftmann@25304
  1168
done
avigad@16775
  1169
lp15@55718
  1170
lemma zero_le_divide_abs_iff [simp]: "(0 \<le> a / abs b) = (0 \<le> a | b = 0)" 
lp15@55718
  1171
by (auto simp: zero_le_divide_iff)
lp15@55718
  1172
lp15@55718
  1173
lemma divide_le_0_abs_iff [simp]: "(a / abs b \<le> 0) = (a \<le> 0 | b = 0)" 
lp15@55718
  1174
by (auto simp: divide_le_0_iff)
lp15@55718
  1175
hoelzl@35579
  1176
lemma field_le_mult_one_interval:
hoelzl@35579
  1177
  assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
hoelzl@35579
  1178
  shows "x \<le> y"
hoelzl@35579
  1179
proof (cases "0 < x")
hoelzl@35579
  1180
  assume "0 < x"
hoelzl@35579
  1181
  thus ?thesis
hoelzl@35579
  1182
    using dense_le_bounded[of 0 1 "y/x"] *
hoelzl@35579
  1183
    unfolding le_divide_eq if_P[OF `0 < x`] by simp
hoelzl@35579
  1184
next
hoelzl@35579
  1185
  assume "\<not>0 < x" hence "x \<le> 0" by simp
hoelzl@35579
  1186
  obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1\<Colon>'a"] by auto
hoelzl@35579
  1187
  hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] `x \<le> 0` by auto
hoelzl@35579
  1188
  also note *[OF s]
hoelzl@35579
  1189
  finally show ?thesis .
hoelzl@35579
  1190
qed
haftmann@35090
  1191
haftmann@36409
  1192
end
haftmann@36409
  1193
haftmann@52435
  1194
code_identifier
haftmann@52435
  1195
  code_module Fields \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  1196
paulson@14265
  1197
end