src/HOL/Fields.thy
author hoelzl
Tue, 27 Aug 2013 14:37:56 +0200
changeset 53215 5e47c31c6f7c
parent 52435 6646bb548c6b
child 53374 a14d2a854c02
permissions -rw-r--r--
renamed typeclass dense_linorder to unbounded_dense_linorder
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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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diff changeset
   147
  by (simp add: divide_inverse)
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   148
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   149
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a / b) = a / (- b)"
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   150
  by (simp add: divide_inverse nonzero_inverse_minus_eq)
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   151
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   152
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a) / (-b) = a / b"
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   153
  by (simp add: divide_inverse nonzero_inverse_minus_eq)
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   154
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   155
lemma divide_minus_left [simp, no_atp]: "(-a) / b = - (a / b)"
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   156
  by (simp add: divide_inverse)
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   157
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   158
lemma diff_divide_distrib: "(a - b) / c = a / c - b / c"
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   159
  by (simp add: diff_minus add_divide_distrib)
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   160
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   161
lemma nonzero_eq_divide_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b"
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   162
proof -
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   163
  assume [simp]: "c \<noteq> 0"
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   164
  have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   165
  also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult_assoc)
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   166
  finally show ?thesis .
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   167
qed
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   168
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   169
lemma nonzero_divide_eq_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> b / c = a \<longleftrightarrow> b = a * c"
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   170
proof -
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   171
  assume [simp]: "c \<noteq> 0"
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   172
  have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   173
  also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult_assoc) 
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   174
  finally show ?thesis .
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   175
qed
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   176
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   177
lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a"
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   178
  by (simp add: divide_inverse mult_assoc)
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   179
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   180
lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   181
  by (drule sym) (simp add: divide_inverse mult_assoc)
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   182
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   183
end
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   184
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   185
class division_ring_inverse_zero = division_ring +
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   186
  assumes inverse_zero [simp]: "inverse 0 = 0"
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   187
begin
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   188
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   189
lemma divide_zero [simp]:
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   190
  "a / 0 = 0"
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   191
  by (simp add: divide_inverse)
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   192
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   193
lemma divide_self_if [simp]:
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   194
  "a / a = (if a = 0 then 0 else 1)"
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   195
  by simp
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   196
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   197
lemma inverse_nonzero_iff_nonzero [simp]:
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   198
  "inverse a = 0 \<longleftrightarrow> a = 0"
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   199
  by rule (fact inverse_zero_imp_zero, simp)
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   200
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   201
lemma inverse_minus_eq [simp]:
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   202
  "inverse (- a) = - inverse a"
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   203
proof cases
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   204
  assume "a=0" thus ?thesis by simp
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   205
next
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   206
  assume "a\<noteq>0" 
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   207
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   208
qed
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   209
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   210
lemma inverse_inverse_eq [simp]:
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   211
  "inverse (inverse a) = a"
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   212
proof cases
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   213
  assume "a=0" thus ?thesis by simp
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   214
next
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   215
  assume "a\<noteq>0" 
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   216
  thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   217
qed
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   218
44680
761f427ef1ab simplify proof
huffman
parents: 44064
diff changeset
   219
lemma inverse_eq_imp_eq:
761f427ef1ab simplify proof
huffman
parents: 44064
diff changeset
   220
  "inverse a = inverse b \<Longrightarrow> a = b"
761f427ef1ab simplify proof
huffman
parents: 44064
diff changeset
   221
  by (drule arg_cong [where f="inverse"], simp)
761f427ef1ab simplify proof
huffman
parents: 44064
diff changeset
   222
761f427ef1ab simplify proof
huffman
parents: 44064
diff changeset
   223
lemma inverse_eq_iff_eq [simp]:
761f427ef1ab simplify proof
huffman
parents: 44064
diff changeset
   224
  "inverse a = inverse b \<longleftrightarrow> a = b"
761f427ef1ab simplify proof
huffman
parents: 44064
diff changeset
   225
  by (force dest!: inverse_eq_imp_eq)
761f427ef1ab simplify proof
huffman
parents: 44064
diff changeset
   226
44064
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   227
end
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   228
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   229
subsection {* Fields *}
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   230
22987
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   231
class field = comm_ring_1 + inverse +
35084
e25eedfc15ce moved constants inverse and divide to Ring.thy
haftmann
parents: 35050
diff changeset
   232
  assumes field_inverse: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
e25eedfc15ce moved constants inverse and divide to Ring.thy
haftmann
parents: 35050
diff changeset
   233
  assumes field_divide_inverse: "a / b = a * inverse b"
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   234
begin
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   235
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   236
subclass division_ring
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28559
diff changeset
   237
proof
22987
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   238
  fix a :: 'a
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   239
  assume "a \<noteq> 0"
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   240
  thus "inverse a * a = 1" by (rule field_inverse)
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   241
  thus "a * inverse a = 1" by (simp only: mult_commute)
35084
e25eedfc15ce moved constants inverse and divide to Ring.thy
haftmann
parents: 35050
diff changeset
   242
next
e25eedfc15ce moved constants inverse and divide to Ring.thy
haftmann
parents: 35050
diff changeset
   243
  fix a b :: 'a
e25eedfc15ce moved constants inverse and divide to Ring.thy
haftmann
parents: 35050
diff changeset
   244
  show "a / b = a * inverse b" by (rule field_divide_inverse)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   245
qed
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   246
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   247
subclass idom ..
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   248
30630
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   249
text{*There is no slick version using division by zero.*}
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   250
lemma inverse_add:
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   251
  "[| a \<noteq> 0;  b \<noteq> 0 |]
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   252
   ==> inverse a + inverse b = (a + b) * inverse a * inverse b"
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   253
by (simp add: division_ring_inverse_add mult_ac)
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   254
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   255
lemma nonzero_mult_divide_mult_cancel_left [simp, no_atp]:
30630
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   256
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/b"
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   257
proof -
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   258
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   259
    by (simp add: divide_inverse nonzero_inverse_mult_distrib)
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   260
  also have "... =  a * inverse b * (inverse c * c)"
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   261
    by (simp only: mult_ac)
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   262
  also have "... =  a * inverse b" by simp
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   263
    finally show ?thesis by (simp add: divide_inverse)
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   264
qed
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   265
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   266
lemma nonzero_mult_divide_mult_cancel_right [simp, no_atp]:
30630
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   267
  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (b * c) = a / b"
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   268
by (simp add: mult_commute [of _ c])
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   269
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   270
lemma times_divide_eq_left [simp]: "(b / c) * a = (b * a) / c"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   271
  by (simp add: divide_inverse mult_ac)
30630
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   272
44921
58eef4843641 tuned proofs
huffman
parents: 44680
diff changeset
   273
text{*It's not obvious whether @{text times_divide_eq} should be
58eef4843641 tuned proofs
huffman
parents: 44680
diff changeset
   274
  simprules or not. Their effect is to gather terms into one big
58eef4843641 tuned proofs
huffman
parents: 44680
diff changeset
   275
  fraction, like a*b*c / x*y*z. The rationale for that is unclear, but
58eef4843641 tuned proofs
huffman
parents: 44680
diff changeset
   276
  many proofs seem to need them.*}
58eef4843641 tuned proofs
huffman
parents: 44680
diff changeset
   277
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   278
lemmas times_divide_eq [no_atp] = times_divide_eq_right times_divide_eq_left
30630
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   279
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   280
lemma add_frac_eq:
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   281
  assumes "y \<noteq> 0" and "z \<noteq> 0"
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   282
  shows "x / y + w / z = (x * z + w * y) / (y * z)"
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   283
proof -
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   284
  have "x / y + w / z = (x * z) / (y * z) + (y * w) / (y * z)"
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   285
    using assms by simp
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   286
  also have "\<dots> = (x * z + y * w) / (y * z)"
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   287
    by (simp only: add_divide_distrib)
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   288
  finally show ?thesis
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   289
    by (simp only: mult_commute)
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   290
qed
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   291
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   292
text{*Special Cancellation Simprules for Division*}
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   293
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   294
lemma nonzero_mult_divide_cancel_right [simp, no_atp]:
30630
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   295
  "b \<noteq> 0 \<Longrightarrow> a * b / b = a"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   296
  using nonzero_mult_divide_mult_cancel_right [of 1 b a] by simp
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   297
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   298
lemma nonzero_mult_divide_cancel_left [simp, no_atp]:
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   299
  "a \<noteq> 0 \<Longrightarrow> a * b / a = b"
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   300
using nonzero_mult_divide_mult_cancel_left [of 1 a b] by simp
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parents: 30242
diff changeset
   301
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parents: 35579
diff changeset
   302
lemma nonzero_divide_mult_cancel_right [simp, no_atp]:
30630
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parents: 30242
diff changeset
   303
  "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> b / (a * b) = 1 / a"
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   304
using nonzero_mult_divide_mult_cancel_right [of a b 1] by simp
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huffman
parents: 30242
diff changeset
   305
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   306
lemma nonzero_divide_mult_cancel_left [simp, no_atp]:
30630
4fbe1401bac2 move field lemmas into class locale context
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parents: 30242
diff changeset
   307
  "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / (a * b) = 1 / b"
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   308
using nonzero_mult_divide_mult_cancel_left [of b a 1] by simp
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   309
35828
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blanchet
parents: 35579
diff changeset
   310
lemma nonzero_mult_divide_mult_cancel_left2 [simp, no_atp]:
30630
4fbe1401bac2 move field lemmas into class locale context
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diff changeset
   311
  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (c * a) / (b * c) = a / b"
4fbe1401bac2 move field lemmas into class locale context
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parents: 30242
diff changeset
   312
using nonzero_mult_divide_mult_cancel_left [of b c a] by (simp add: mult_ac)
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parents: 30242
diff changeset
   313
35828
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parents: 35579
diff changeset
   314
lemma nonzero_mult_divide_mult_cancel_right2 [simp, no_atp]:
30630
4fbe1401bac2 move field lemmas into class locale context
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parents: 30242
diff changeset
   315
  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (c * b) = a / b"
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   316
using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: mult_ac)
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huffman
parents: 30242
diff changeset
   317
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
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diff changeset
   318
lemma add_divide_eq_iff [field_simps]:
30630
4fbe1401bac2 move field lemmas into class locale context
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diff changeset
   319
  "z \<noteq> 0 \<Longrightarrow> x + y / z = (z * x + y) / z"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
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diff changeset
   320
  by (simp add: add_divide_distrib)
30630
4fbe1401bac2 move field lemmas into class locale context
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parents: 30242
diff changeset
   321
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
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   322
lemma divide_add_eq_iff [field_simps]:
30630
4fbe1401bac2 move field lemmas into class locale context
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diff changeset
   323
  "z \<noteq> 0 \<Longrightarrow> x / z + y = (x + z * y) / z"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
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diff changeset
   324
  by (simp add: add_divide_distrib)
30630
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parents: 30242
diff changeset
   325
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   326
lemma diff_divide_eq_iff [field_simps]:
30630
4fbe1401bac2 move field lemmas into class locale context
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diff changeset
   327
  "z \<noteq> 0 \<Longrightarrow> x - y / z = (z * x - y) / z"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
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parents: 35828
diff changeset
   328
  by (simp add: diff_divide_distrib)
30630
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   329
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   330
lemma divide_diff_eq_iff [field_simps]:
30630
4fbe1401bac2 move field lemmas into class locale context
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parents: 30242
diff changeset
   331
  "z \<noteq> 0 \<Longrightarrow> x / z - y = (x - z * y) / z"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   332
  by (simp add: diff_divide_distrib)
30630
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   333
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   334
lemma diff_frac_eq:
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   335
  "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y - w / z = (x * z - w * y) / (y * z)"
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   336
  by (simp add: field_simps)
30630
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   337
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   338
lemma frac_eq_eq:
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   339
  "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)"
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   340
  by (simp add: field_simps)
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   341
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   342
end
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   343
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   344
class field_inverse_zero = field +
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   345
  assumes field_inverse_zero: "inverse 0 = 0"
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   346
begin
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   347
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   348
subclass division_ring_inverse_zero proof
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   349
qed (fact field_inverse_zero)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   350
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   351
text{*This version builds in division by zero while also re-orienting
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   352
      the right-hand side.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   353
lemma inverse_mult_distrib [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
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diff changeset
   354
  "inverse (a * b) = inverse a * inverse b"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   355
proof cases
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   356
  assume "a \<noteq> 0 & b \<noteq> 0" 
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   357
  thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_ac)
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   358
next
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   359
  assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   360
  thus ?thesis by force
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   361
qed
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   362
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   363
lemma inverse_divide [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   364
  "inverse (a / b) = b / a"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   365
  by (simp add: divide_inverse mult_commute)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   366
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   367
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   368
text {* Calculations with fractions *}
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   369
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
   370
text{* There is a whole bunch of simp-rules just for class @{text
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
   371
field} but none for class @{text field} and @{text nonzero_divides}
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
   372
because the latter are covered by a simproc. *}
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
   373
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
   374
lemma mult_divide_mult_cancel_left:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   375
  "c \<noteq> 0 \<Longrightarrow> (c * a) / (c * b) = a / b"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   376
apply (cases "b = 0")
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35090
diff changeset
   377
apply simp_all
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   378
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   379
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
   380
lemma mult_divide_mult_cancel_right:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   381
  "c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   382
apply (cases "b = 0")
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35090
diff changeset
   383
apply simp_all
14321
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   384
done
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
   385
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   386
lemma divide_divide_eq_right [simp, no_atp]:
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   387
  "a / (b / c) = (a * c) / b"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   388
  by (simp add: divide_inverse mult_ac)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   389
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   390
lemma divide_divide_eq_left [simp, no_atp]:
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   391
  "(a / b) / c = a / (b * c)"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   392
  by (simp add: divide_inverse mult_assoc)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   393
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   394
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   395
text {*Special Cancellation Simprules for Division*}
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   396
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   397
lemma mult_divide_mult_cancel_left_if [simp,no_atp]:
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   398
  shows "(c * a) / (c * b) = (if c = 0 then 0 else a / b)"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   399
  by (simp add: mult_divide_mult_cancel_left)
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
   400
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   401
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   402
text {* Division and Unary Minus *}
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   403
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   404
lemma minus_divide_right:
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   405
  "- (a / b) = a / - b"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   406
  by (simp add: divide_inverse)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   407
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   408
lemma divide_minus_right [simp, no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   409
  "a / - b = - (a / b)"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   410
  by (simp add: divide_inverse)
30630
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   411
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   412
lemma minus_divide_divide:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   413
  "(- a) / (- b) = a / b"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   414
apply (cases "b=0", simp) 
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   415
apply (simp add: nonzero_minus_divide_divide) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   416
done
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   417
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   418
lemma eq_divide_eq:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   419
  "a = b / c \<longleftrightarrow> (if c \<noteq> 0 then a * c = b else a = 0)"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   420
  by (simp add: nonzero_eq_divide_eq)
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   421
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   422
lemma divide_eq_eq:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   423
  "b / c = a \<longleftrightarrow> (if c \<noteq> 0 then b = a * c else a = 0)"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   424
  by (force simp add: nonzero_divide_eq_eq)
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   425
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   426
lemma inverse_eq_1_iff [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   427
  "inverse x = 1 \<longleftrightarrow> x = 1"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   428
  by (insert inverse_eq_iff_eq [of x 1], simp) 
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   429
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   430
lemma divide_eq_0_iff [simp, no_atp]:
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   431
  "a / b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   432
  by (simp add: divide_inverse)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   433
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   434
lemma divide_cancel_right [simp, no_atp]:
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   435
  "a / c = b / c \<longleftrightarrow> c = 0 \<or> a = b"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   436
  apply (cases "c=0", simp)
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   437
  apply (simp add: divide_inverse)
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   438
  done
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   439
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   440
lemma divide_cancel_left [simp, no_atp]:
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   441
  "c / a = c / b \<longleftrightarrow> c = 0 \<or> a = b" 
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   442
  apply (cases "c=0", simp)
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   443
  apply (simp add: divide_inverse)
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   444
  done
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   445
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   446
lemma divide_eq_1_iff [simp, no_atp]:
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   447
  "a / b = 1 \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   448
  apply (cases "b=0", simp)
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   449
  apply (simp add: right_inverse_eq)
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   450
  done
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   451
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   452
lemma one_eq_divide_iff [simp, no_atp]:
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   453
  "1 = a / b \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   454
  by (simp add: eq_commute [of 1])
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   455
36719
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36425
diff changeset
   456
lemma times_divide_times_eq:
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36425
diff changeset
   457
  "(x / y) * (z / w) = (x * z) / (y * w)"
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36425
diff changeset
   458
  by simp
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36425
diff changeset
   459
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36425
diff changeset
   460
lemma add_frac_num:
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36425
diff changeset
   461
  "y \<noteq> 0 \<Longrightarrow> x / y + z = (x + z * y) / y"
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36425
diff changeset
   462
  by (simp add: add_divide_distrib)
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36425
diff changeset
   463
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36425
diff changeset
   464
lemma add_num_frac:
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36425
diff changeset
   465
  "y \<noteq> 0 \<Longrightarrow> z + x / y = (x + z * y) / y"
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36425
diff changeset
   466
  by (simp add: add_divide_distrib add.commute)
d396f6f63d94 moved some lemmas from Groebner_Basis here
haftmann
parents: 36425
diff changeset
   467
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   468
end
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   469
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   470
44064
5bce8ff0d9ae moved division ring stuff from Rings.thy to Fields.thy
huffman
parents: 42904
diff changeset
   471
subsection {* Ordered fields *}
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   472
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   473
class linordered_field = field + linordered_idom
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   474
begin
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   475
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   476
lemma positive_imp_inverse_positive: 
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   477
  assumes a_gt_0: "0 < a" 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   478
  shows "0 < inverse a"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   479
proof -
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   480
  have "0 < a * inverse a" 
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   481
    by (simp add: a_gt_0 [THEN less_imp_not_eq2])
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   482
  thus "0 < inverse a" 
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   483
    by (simp add: a_gt_0 [THEN less_not_sym] zero_less_mult_iff)
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   484
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   485
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   486
lemma negative_imp_inverse_negative:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   487
  "a < 0 \<Longrightarrow> inverse a < 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   488
  by (insert positive_imp_inverse_positive [of "-a"], 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   489
    simp add: nonzero_inverse_minus_eq less_imp_not_eq)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   490
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   491
lemma inverse_le_imp_le:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   492
  assumes invle: "inverse a \<le> inverse b" and apos: "0 < a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   493
  shows "b \<le> a"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   494
proof (rule classical)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   495
  assume "~ b \<le> a"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   496
  hence "a < b"  by (simp add: linorder_not_le)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   497
  hence bpos: "0 < b"  by (blast intro: apos less_trans)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   498
  hence "a * inverse a \<le> a * inverse b"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   499
    by (simp add: apos invle less_imp_le mult_left_mono)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   500
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   501
    by (simp add: bpos less_imp_le mult_right_mono)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   502
  thus "b \<le> a"  by (simp add: mult_assoc apos bpos less_imp_not_eq2)
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   503
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   504
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   505
lemma inverse_positive_imp_positive:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   506
  assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   507
  shows "0 < a"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   508
proof -
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   509
  have "0 < inverse (inverse a)"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   510
    using inv_gt_0 by (rule positive_imp_inverse_positive)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   511
  thus "0 < a"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   512
    using nz by (simp add: nonzero_inverse_inverse_eq)
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   513
qed
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   514
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   515
lemma inverse_negative_imp_negative:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   516
  assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   517
  shows "a < 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   518
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   519
  have "inverse (inverse a) < 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   520
    using inv_less_0 by (rule negative_imp_inverse_negative)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   521
  thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   522
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   523
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   524
lemma linordered_field_no_lb:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   525
  "\<forall>x. \<exists>y. y < x"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   526
proof
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   527
  fix x::'a
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   528
  have m1: "- (1::'a) < 0" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   529
  from add_strict_right_mono[OF m1, where c=x] 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   530
  have "(- 1) + x < x" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   531
  thus "\<exists>y. y < x" by blast
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   532
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   533
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   534
lemma linordered_field_no_ub:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   535
  "\<forall> x. \<exists>y. y > x"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   536
proof
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   537
  fix x::'a
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   538
  have m1: " (1::'a) > 0" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   539
  from add_strict_right_mono[OF m1, where c=x] 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   540
  have "1 + x > x" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   541
  thus "\<exists>y. y > x" by blast
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   542
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   543
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   544
lemma less_imp_inverse_less:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   545
  assumes less: "a < b" and apos:  "0 < a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   546
  shows "inverse b < inverse a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   547
proof (rule ccontr)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   548
  assume "~ inverse b < inverse a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   549
  hence "inverse a \<le> inverse b" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   550
  hence "~ (a < b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   551
    by (simp add: not_less inverse_le_imp_le [OF _ apos])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   552
  thus False by (rule notE [OF _ less])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   553
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   554
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   555
lemma inverse_less_imp_less:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   556
  "inverse a < inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b < a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   557
apply (simp add: less_le [of "inverse a"] less_le [of "b"])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   558
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   559
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   560
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   561
text{*Both premises are essential. Consider -1 and 1.*}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   562
lemma inverse_less_iff_less [simp,no_atp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   563
  "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   564
  by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   565
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   566
lemma le_imp_inverse_le:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   567
  "a \<le> b \<Longrightarrow> 0 < a \<Longrightarrow> inverse b \<le> inverse a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   568
  by (force simp add: le_less less_imp_inverse_less)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   569
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   570
lemma inverse_le_iff_le [simp,no_atp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   571
  "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   572
  by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   573
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   574
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   575
text{*These results refer to both operands being negative.  The opposite-sign
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   576
case is trivial, since inverse preserves signs.*}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   577
lemma inverse_le_imp_le_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   578
  "inverse a \<le> inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   579
apply (rule classical) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   580
apply (subgoal_tac "a < 0") 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   581
 prefer 2 apply force
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   582
apply (insert inverse_le_imp_le [of "-b" "-a"])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   583
apply (simp add: nonzero_inverse_minus_eq) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   584
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   585
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   586
lemma less_imp_inverse_less_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   587
   "a < b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b < inverse a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   588
apply (subgoal_tac "a < 0") 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   589
 prefer 2 apply (blast intro: less_trans) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   590
apply (insert less_imp_inverse_less [of "-b" "-a"])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   591
apply (simp add: nonzero_inverse_minus_eq) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   592
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   593
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   594
lemma inverse_less_imp_less_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   595
   "inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   596
apply (rule classical) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   597
apply (subgoal_tac "a < 0") 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   598
 prefer 2
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   599
 apply force
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   600
apply (insert inverse_less_imp_less [of "-b" "-a"])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   601
apply (simp add: nonzero_inverse_minus_eq) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   602
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   603
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   604
lemma inverse_less_iff_less_neg [simp,no_atp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   605
  "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   606
apply (insert inverse_less_iff_less [of "-b" "-a"])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   607
apply (simp del: inverse_less_iff_less 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   608
            add: nonzero_inverse_minus_eq)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   609
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   610
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   611
lemma le_imp_inverse_le_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   612
  "a \<le> b \<Longrightarrow> b < 0 ==> inverse b \<le> inverse a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   613
  by (force simp add: le_less less_imp_inverse_less_neg)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   614
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   615
lemma inverse_le_iff_le_neg [simp,no_atp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   616
  "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   617
  by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   618
36774
9e444b09fbef add lemmas one_less_inverse and one_le_inverse
huffman
parents: 36719
diff changeset
   619
lemma one_less_inverse:
9e444b09fbef add lemmas one_less_inverse and one_le_inverse
huffman
parents: 36719
diff changeset
   620
  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> 1 < inverse a"
9e444b09fbef add lemmas one_less_inverse and one_le_inverse
huffman
parents: 36719
diff changeset
   621
  using less_imp_inverse_less [of a 1, unfolded inverse_1] .
9e444b09fbef add lemmas one_less_inverse and one_le_inverse
huffman
parents: 36719
diff changeset
   622
9e444b09fbef add lemmas one_less_inverse and one_le_inverse
huffman
parents: 36719
diff changeset
   623
lemma one_le_inverse:
9e444b09fbef add lemmas one_less_inverse and one_le_inverse
huffman
parents: 36719
diff changeset
   624
  "0 < a \<Longrightarrow> a \<le> 1 \<Longrightarrow> 1 \<le> inverse a"
9e444b09fbef add lemmas one_less_inverse and one_le_inverse
huffman
parents: 36719
diff changeset
   625
  using le_imp_inverse_le [of a 1, unfolded inverse_1] .
9e444b09fbef add lemmas one_less_inverse and one_le_inverse
huffman
parents: 36719
diff changeset
   626
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   627
lemma pos_le_divide_eq [field_simps]: "0 < c ==> (a \<le> b/c) = (a*c \<le> b)"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   628
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   629
  assume less: "0<c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   630
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   631
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   632
  also have "... = (a*c \<le> b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   633
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   634
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   635
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   636
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   637
lemma neg_le_divide_eq [field_simps]: "c < 0 ==> (a \<le> b/c) = (b \<le> a*c)"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   638
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   639
  assume less: "c<0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   640
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   641
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   642
  also have "... = (b \<le> a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   643
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   644
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   645
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   646
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   647
lemma pos_less_divide_eq [field_simps]:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   648
     "0 < c ==> (a < b/c) = (a*c < b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   649
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   650
  assume less: "0<c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   651
  hence "(a < b/c) = (a*c < (b/c)*c)"
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   652
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   653
  also have "... = (a*c < b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   654
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   655
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   656
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   657
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   658
lemma neg_less_divide_eq [field_simps]:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   659
 "c < 0 ==> (a < b/c) = (b < a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   660
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   661
  assume less: "c<0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   662
  hence "(a < b/c) = ((b/c)*c < a*c)"
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   663
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   664
  also have "... = (b < a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   665
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   666
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   667
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   668
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   669
lemma pos_divide_less_eq [field_simps]:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   670
     "0 < c ==> (b/c < a) = (b < a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   671
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   672
  assume less: "0<c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   673
  hence "(b/c < a) = ((b/c)*c < a*c)"
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   674
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   675
  also have "... = (b < a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   676
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   677
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   678
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   679
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   680
lemma neg_divide_less_eq [field_simps]:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   681
 "c < 0 ==> (b/c < a) = (a*c < b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   682
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   683
  assume less: "c<0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   684
  hence "(b/c < a) = (a*c < (b/c)*c)"
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   685
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   686
  also have "... = (a*c < b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   687
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   688
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   689
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   690
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   691
lemma pos_divide_le_eq [field_simps]: "0 < c ==> (b/c \<le> a) = (b \<le> a*c)"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   692
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   693
  assume less: "0<c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   694
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   695
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   696
  also have "... = (b \<le> a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   697
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   698
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   699
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   700
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   701
lemma neg_divide_le_eq [field_simps]: "c < 0 ==> (b/c \<le> a) = (a*c \<le> b)"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   702
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   703
  assume less: "c<0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   704
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   705
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   706
  also have "... = (a*c \<le> b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   707
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   708
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   709
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   710
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   711
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   712
of positivity/negativity needed for @{text field_simps}. Have not added @{text
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   713
sign_simps} to @{text field_simps} because the former can lead to case
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   714
explosions. *}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   715
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   716
lemmas sign_simps [no_atp] = algebra_simps
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   717
  zero_less_mult_iff mult_less_0_iff
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   718
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   719
lemmas (in -) sign_simps [no_atp] = algebra_simps
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   720
  zero_less_mult_iff mult_less_0_iff
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   721
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   722
(* Only works once linear arithmetic is installed:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   723
text{*An example:*}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   724
lemma fixes a b c d e f :: "'a::linordered_field"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   725
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   726
 ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   727
 ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   728
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   729
 prefer 2 apply(simp add:sign_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   730
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   731
 prefer 2 apply(simp add:sign_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   732
apply(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   733
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   734
*)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   735
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   736
lemma divide_pos_pos:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   737
  "0 < x ==> 0 < y ==> 0 < x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   738
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   739
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   740
lemma divide_nonneg_pos:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   741
  "0 <= x ==> 0 < y ==> 0 <= x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   742
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   743
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   744
lemma divide_neg_pos:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   745
  "x < 0 ==> 0 < y ==> x / y < 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   746
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   747
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   748
lemma divide_nonpos_pos:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   749
  "x <= 0 ==> 0 < y ==> x / y <= 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   750
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   751
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   752
lemma divide_pos_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   753
  "0 < x ==> y < 0 ==> x / y < 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   754
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   755
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   756
lemma divide_nonneg_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   757
  "0 <= x ==> y < 0 ==> x / y <= 0" 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   758
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   759
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   760
lemma divide_neg_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   761
  "x < 0 ==> y < 0 ==> 0 < x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   762
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   763
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   764
lemma divide_nonpos_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   765
  "x <= 0 ==> y < 0 ==> 0 <= x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   766
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   767
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   768
lemma divide_strict_right_mono:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   769
     "[|a < b; 0 < c|] ==> a / c < b / c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   770
by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   771
              positive_imp_inverse_positive)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   772
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   773
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   774
lemma divide_strict_right_mono_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   775
     "[|b < a; c < 0|] ==> a / c < b / c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   776
apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   777
apply (simp add: less_imp_not_eq nonzero_minus_divide_right [symmetric])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   778
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   779
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   780
text{*The last premise ensures that @{term a} and @{term b} 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   781
      have the same sign*}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   782
lemma divide_strict_left_mono:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   783
  "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / b"
44921
58eef4843641 tuned proofs
huffman
parents: 44680
diff changeset
   784
  by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   785
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   786
lemma divide_left_mono:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   787
  "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / b"
44921
58eef4843641 tuned proofs
huffman
parents: 44680
diff changeset
   788
  by (auto simp: field_simps zero_less_mult_iff mult_right_mono)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   789
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   790
lemma divide_strict_left_mono_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   791
  "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / b"
44921
58eef4843641 tuned proofs
huffman
parents: 44680
diff changeset
   792
  by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono_neg)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   793
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   794
lemma mult_imp_div_pos_le: "0 < y ==> x <= z * y ==>
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   795
    x / y <= z"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   796
by (subst pos_divide_le_eq, assumption+)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   797
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   798
lemma mult_imp_le_div_pos: "0 < y ==> z * y <= x ==>
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   799
    z <= x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   800
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   801
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   802
lemma mult_imp_div_pos_less: "0 < y ==> x < z * y ==>
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   803
    x / y < z"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   804
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   805
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   806
lemma mult_imp_less_div_pos: "0 < y ==> z * y < x ==>
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   807
    z < x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   808
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   809
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   810
lemma frac_le: "0 <= x ==> 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   811
    x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   812
  apply (rule mult_imp_div_pos_le)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   813
  apply simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   814
  apply (subst times_divide_eq_left)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   815
  apply (rule mult_imp_le_div_pos, assumption)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   816
  apply (rule mult_mono)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   817
  apply simp_all
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   818
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   819
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   820
lemma frac_less: "0 <= x ==> 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   821
    x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   822
  apply (rule mult_imp_div_pos_less)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   823
  apply simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   824
  apply (subst times_divide_eq_left)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   825
  apply (rule mult_imp_less_div_pos, assumption)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   826
  apply (erule mult_less_le_imp_less)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   827
  apply simp_all
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   828
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   829
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   830
lemma frac_less2: "0 < x ==> 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   831
    x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   832
  apply (rule mult_imp_div_pos_less)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   833
  apply simp_all
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   834
  apply (rule mult_imp_less_div_pos, assumption)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   835
  apply (erule mult_le_less_imp_less)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   836
  apply simp_all
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   837
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   838
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   839
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   840
by (simp add: field_simps zero_less_two)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   841
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   842
lemma gt_half_sum: "a < b ==> (a+b)/(1+1) < b"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   843
by (simp add: field_simps zero_less_two)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   844
53215
5e47c31c6f7c renamed typeclass dense_linorder to unbounded_dense_linorder
hoelzl
parents: 52435
diff changeset
   845
subclass unbounded_dense_linorder
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   846
proof
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   847
  fix x y :: 'a
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   848
  from less_add_one show "\<exists>y. x < y" .. 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   849
  from less_add_one have "x + (- 1) < (x + 1) + (- 1)" by (rule add_strict_right_mono)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   850
  then have "x - 1 < x + 1 - 1" by (simp only: diff_minus [symmetric])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   851
  then have "x - 1 < x" by (simp add: algebra_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   852
  then show "\<exists>y. y < x" ..
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   853
  show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   854
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   855
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   856
lemma nonzero_abs_inverse:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   857
     "a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   858
apply (auto simp add: neq_iff abs_if nonzero_inverse_minus_eq 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   859
                      negative_imp_inverse_negative)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   860
apply (blast intro: positive_imp_inverse_positive elim: less_asym) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   861
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   862
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   863
lemma nonzero_abs_divide:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   864
     "b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   865
  by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   866
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   867
lemma field_le_epsilon:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   868
  assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   869
  shows "x \<le> y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   870
proof (rule dense_le)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   871
  fix t assume "t < x"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   872
  hence "0 < x - t" by (simp add: less_diff_eq)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   873
  from e [OF this] have "x + 0 \<le> x + (y - t)" by (simp add: algebra_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   874
  then have "0 \<le> y - t" by (simp only: add_le_cancel_left)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   875
  then show "t \<le> y" by (simp add: algebra_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   876
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   877
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   878
end
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   879
36414
a19ba9bbc8dc tuned class linordered_field_inverse_zero
haftmann
parents: 36409
diff changeset
   880
class linordered_field_inverse_zero = linordered_field + field_inverse_zero
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   881
begin
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   882
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   883
lemma le_divide_eq:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   884
  "(a \<le> b/c) = 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   885
   (if 0 < c then a*c \<le> b
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   886
             else if c < 0 then b \<le> a*c
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   887
             else  a \<le> 0)"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   888
apply (cases "c=0", simp) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   889
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   890
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   891
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   892
lemma inverse_positive_iff_positive [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   893
  "(0 < inverse a) = (0 < a)"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   894
apply (cases "a = 0", simp)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   895
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   896
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   897
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   898
lemma inverse_negative_iff_negative [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   899
  "(inverse a < 0) = (a < 0)"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   900
apply (cases "a = 0", simp)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   901
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   902
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   903
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   904
lemma inverse_nonnegative_iff_nonnegative [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   905
  "0 \<le> inverse a \<longleftrightarrow> 0 \<le> a"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   906
  by (simp add: not_less [symmetric])
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   907
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   908
lemma inverse_nonpositive_iff_nonpositive [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   909
  "inverse a \<le> 0 \<longleftrightarrow> a \<le> 0"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   910
  by (simp add: not_less [symmetric])
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   911
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   912
lemma one_less_inverse_iff:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   913
  "1 < inverse x \<longleftrightarrow> 0 < x \<and> x < 1"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   914
proof cases
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   915
  assume "0 < x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   916
    with inverse_less_iff_less [OF zero_less_one, of x]
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   917
    show ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   918
next
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   919
  assume notless: "~ (0 < x)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   920
  have "~ (1 < inverse x)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   921
  proof
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   922
    assume "1 < inverse x"
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   923
    also with notless have "... \<le> 0" by simp
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   924
    also have "... < 1" by (rule zero_less_one) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   925
    finally show False by auto
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   926
  qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   927
  with notless show ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   928
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   929
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   930
lemma one_le_inverse_iff:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   931
  "1 \<le> inverse x \<longleftrightarrow> 0 < x \<and> x \<le> 1"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   932
proof (cases "x = 1")
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   933
  case True then show ?thesis by simp
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   934
next
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   935
  case False then have "inverse x \<noteq> 1" by simp
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   936
  then have "1 \<noteq> inverse x" by blast
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   937
  then have "1 \<le> inverse x \<longleftrightarrow> 1 < inverse x" by (simp add: le_less)
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   938
  with False show ?thesis by (auto simp add: one_less_inverse_iff)
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   939
qed
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   940
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   941
lemma inverse_less_1_iff:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   942
  "inverse x < 1 \<longleftrightarrow> x \<le> 0 \<or> 1 < x"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   943
  by (simp add: not_le [symmetric] one_le_inverse_iff) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   944
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   945
lemma inverse_le_1_iff:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   946
  "inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   947
  by (simp add: not_less [symmetric] one_less_inverse_iff) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   948
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   949
lemma divide_le_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   950
  "(b/c \<le> a) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   951
   (if 0 < c then b \<le> a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   952
             else if c < 0 then a*c \<le> b
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   953
             else 0 \<le> a)"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   954
apply (cases "c=0", simp) 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   955
apply (force simp add: pos_divide_le_eq neg_divide_le_eq) 
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   956
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   957
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   958
lemma less_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   959
  "(a < b/c) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   960
   (if 0 < c then a*c < b
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   961
             else if c < 0 then b < a*c
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   962
             else  a < 0)"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   963
apply (cases "c=0", simp) 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   964
apply (force simp add: pos_less_divide_eq neg_less_divide_eq) 
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   965
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   966
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   967
lemma divide_less_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   968
  "(b/c < a) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   969
   (if 0 < c then b < a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   970
             else if c < 0 then a*c < b
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   971
             else 0 < a)"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   972
apply (cases "c=0", simp) 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   973
apply (force simp add: pos_divide_less_eq neg_divide_less_eq)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   974
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   975
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   976
text {*Division and Signs*}
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   977
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   978
lemma zero_less_divide_iff:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   979
     "(0 < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   980
by (simp add: divide_inverse zero_less_mult_iff)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   981
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   982
lemma divide_less_0_iff:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   983
     "(a/b < 0) = 
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   984
      (0 < a & b < 0 | a < 0 & 0 < b)"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   985
by (simp add: divide_inverse mult_less_0_iff)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   986
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   987
lemma zero_le_divide_iff:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   988
     "(0 \<le> a/b) =
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   989
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   990
by (simp add: divide_inverse zero_le_mult_iff)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   991
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   992
lemma divide_le_0_iff:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   993
     "(a/b \<le> 0) =
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   994
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   995
by (simp add: divide_inverse mult_le_0_iff)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   996
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   997
text {* Division and the Number One *}
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   998
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   999
text{*Simplify expressions equated with 1*}
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1000
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
  1001
lemma zero_eq_1_divide_iff [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1002
     "(0 = 1/a) = (a = 0)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1003
apply (cases "a=0", simp)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1004
apply (auto simp add: nonzero_eq_divide_eq)
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1005
done
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1006
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
  1007
lemma one_divide_eq_0_iff [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1008
     "(1/a = 0) = (a = 0)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1009
apply (cases "a=0", simp)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1010
apply (insert zero_neq_one [THEN not_sym])
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1011
apply (auto simp add: nonzero_divide_eq_eq)
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1012
done
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1013
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1014
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
36423
63fc238a7430 got rid of [simplified]
haftmann
parents: 36414
diff changeset
  1015
63fc238a7430 got rid of [simplified]
haftmann
parents: 36414
diff changeset
  1016
lemma zero_le_divide_1_iff [simp, no_atp]:
63fc238a7430 got rid of [simplified]
haftmann
parents: 36414
diff changeset
  1017
  "0 \<le> 1 / a \<longleftrightarrow> 0 \<le> a"
63fc238a7430 got rid of [simplified]
haftmann
parents: 36414
diff changeset
  1018
  by (simp add: zero_le_divide_iff)
17085
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1019
36423
63fc238a7430 got rid of [simplified]
haftmann
parents: 36414
diff changeset
  1020
lemma zero_less_divide_1_iff [simp, no_atp]:
63fc238a7430 got rid of [simplified]
haftmann
parents: 36414
diff changeset
  1021
  "0 < 1 / a \<longleftrightarrow> 0 < a"
63fc238a7430 got rid of [simplified]
haftmann
parents: 36414
diff changeset
  1022
  by (simp add: zero_less_divide_iff)
63fc238a7430 got rid of [simplified]
haftmann
parents: 36414
diff changeset
  1023
63fc238a7430 got rid of [simplified]
haftmann
parents: 36414
diff changeset
  1024
lemma divide_le_0_1_iff [simp, no_atp]:
63fc238a7430 got rid of [simplified]
haftmann
parents: 36414
diff changeset
  1025
  "1 / a \<le> 0 \<longleftrightarrow> a \<le> 0"
63fc238a7430 got rid of [simplified]
haftmann
parents: 36414
diff changeset
  1026
  by (simp add: divide_le_0_iff)
63fc238a7430 got rid of [simplified]
haftmann
parents: 36414
diff changeset
  1027
63fc238a7430 got rid of [simplified]
haftmann
parents: 36414
diff changeset
  1028
lemma divide_less_0_1_iff [simp, no_atp]:
63fc238a7430 got rid of [simplified]
haftmann
parents: 36414
diff changeset
  1029
  "1 / a < 0 \<longleftrightarrow> a < 0"
63fc238a7430 got rid of [simplified]
haftmann
parents: 36414
diff changeset
  1030
  by (simp add: divide_less_0_iff)
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
  1031
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1032
lemma divide_right_mono:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1033
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/c"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1034
by (force simp add: divide_strict_right_mono le_less)
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1035
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1036
lemma divide_right_mono_neg: "a <= b 
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1037
    ==> c <= 0 ==> b / c <= a / c"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1038
apply (drule divide_right_mono [of _ _ "- c"])
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1039
apply auto
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1040
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1041
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1042
lemma divide_left_mono_neg: "a <= b 
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1043
    ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1044
  apply (drule divide_left_mono [of _ _ "- c"])
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1045
  apply (auto simp add: mult_commute)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1046
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1047
42904
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1048
lemma inverse_le_iff:
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1049
  "inverse a \<le> inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b \<le> a) \<and> (a * b \<le> 0 \<longrightarrow> a \<le> b)"
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1050
proof -
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1051
  { assume "a < 0"
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1052
    then have "inverse a < 0" by simp
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1053
    moreover assume "0 < b"
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1054
    then have "0 < inverse b" by simp
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1055
    ultimately have "inverse a < inverse b" by (rule less_trans)
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1056
    then have "inverse a \<le> inverse b" by simp }
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1057
  moreover
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1058
  { assume "b < 0"
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1059
    then have "inverse b < 0" by simp
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1060
    moreover assume "0 < a"
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1061
    then have "0 < inverse a" by simp
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1062
    ultimately have "inverse b < inverse a" by (rule less_trans)
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1063
    then have "\<not> inverse a \<le> inverse b" by simp }
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1064
  ultimately show ?thesis
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1065
    by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1066
       (auto simp: not_less zero_less_mult_iff mult_le_0_iff)
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1067
qed
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1068
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1069
lemma inverse_less_iff:
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1070
  "inverse a < inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b < a) \<and> (a * b \<le> 0 \<longrightarrow> a < b)"
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1071
  by (subst less_le) (auto simp: inverse_le_iff)
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1072
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1073
lemma divide_le_cancel:
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1074
  "a / c \<le> b / c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1075
  by (simp add: divide_inverse mult_le_cancel_right)
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1076
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1077
lemma divide_less_cancel:
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1078
  "a / c < b / c \<longleftrightarrow> (0 < c \<longrightarrow> a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0"
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1079
  by (auto simp add: divide_inverse mult_less_cancel_right)
4aedcff42de6 add divide_.._cancel, inverse_.._iff
hoelzl
parents: 36774
diff changeset
  1080
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1081
text{*Simplify quotients that are compared with the value 1.*}
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1082
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
  1083
lemma le_divide_eq_1 [no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1084
  "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1085
by (auto simp add: le_divide_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1086
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
  1087
lemma divide_le_eq_1 [no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1088
  "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1089
by (auto simp add: divide_le_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1090
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
  1091
lemma less_divide_eq_1 [no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1092
  "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1093
by (auto simp add: less_divide_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1094
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
  1095
lemma divide_less_eq_1 [no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1096
  "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1097
by (auto simp add: divide_less_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1098
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
  1099
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1100
text {*Conditional Simplification Rules: No Case Splits*}
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1101
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
  1102
lemma le_divide_eq_1_pos [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1103
  "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1104
by (auto simp add: le_divide_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1105
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
  1106
lemma le_divide_eq_1_neg [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1107
  "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1108
by (auto simp add: le_divide_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1109
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
  1110
lemma divide_le_eq_1_pos [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1111
  "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1112
by (auto simp add: divide_le_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1113
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
  1114
lemma divide_le_eq_1_neg [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1115
  "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1116
by (auto simp add: divide_le_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1117
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
  1118
lemma less_divide_eq_1_pos [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1119
  "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1120
by (auto simp add: less_divide_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1121
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
  1122
lemma less_divide_eq_1_neg [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1123
  "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1124
by (auto simp add: less_divide_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1125
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
  1126
lemma divide_less_eq_1_pos [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1127
  "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
18649
bb99c2e705ca tidied, and added missing thm divide_less_eq_1_neg
paulson
parents: 18623
diff changeset
  1128
by (auto simp add: divide_less_eq)
bb99c2e705ca tidied, and added missing thm divide_less_eq_1_neg
paulson
parents: 18623
diff changeset
  1129
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
  1130
lemma divide_less_eq_1_neg [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1131
  "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1132
by (auto simp add: divide_less_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1133
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
  1134
lemma eq_divide_eq_1 [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1135
  "(1 = b/a) = ((a \<noteq> 0 & a = b))"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1136
by (auto simp add: eq_divide_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1137
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
  1138
lemma divide_eq_eq_1 [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1139
  "(b/a = 1) = ((a \<noteq> 0 & a = b))"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1140
by (auto simp add: divide_eq_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1141
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1142
lemma abs_inverse [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1143
     "\<bar>inverse a\<bar> = 
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1144
      inverse \<bar>a\<bar>"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
  1145
apply (cases "a=0", simp) 
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1146
apply (simp add: nonzero_abs_inverse) 
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1147
done
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1148
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1149
lemma abs_divide [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1150
     "\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
  1151
apply (cases "b=0", simp) 
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1152
apply (simp add: nonzero_abs_divide) 
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1153
done
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1154
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1155
lemma abs_div_pos: "0 < y ==> 
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
  1156
    \<bar>x\<bar> / y = \<bar>x / y\<bar>"
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1157
  apply (subst abs_divide)
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1158
  apply (simp add: order_less_imp_le)
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1159
done
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1160
35579
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
  1161
lemma field_le_mult_one_interval:
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
  1162
  assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
  1163
  shows "x \<le> y"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
  1164
proof (cases "0 < x")
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
  1165
  assume "0 < x"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
  1166
  thus ?thesis
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
  1167
    using dense_le_bounded[of 0 1 "y/x"] *
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
  1168
    unfolding le_divide_eq if_P[OF `0 < x`] by simp
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
  1169
next
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
  1170
  assume "\<not>0 < x" hence "x \<le> 0" by simp
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
  1171
  obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1\<Colon>'a"] by auto
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
  1172
  hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] `x \<le> 0` by auto
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
  1173
  also note *[OF s]
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
  1174
  finally show ?thesis .
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
  1175
qed
35090
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1176
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1177
end
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
  1178
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 44921
diff changeset
  1179
code_identifier
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 44921
diff changeset
  1180
  code_module Fields \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
  1181
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
  1182
end