src/HOL/Ring_and_Field.thy
author paulson
Thu, 01 Apr 2004 10:54:32 +0200
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(*  Title:   HOL/Ring_and_Field.thy
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    ID:      $Id$
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    Author:  Gertrud Bauer and Markus Wenzel, TU Muenchen
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             Lawrence C Paulson, University of Cambridge
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    License: GPL (GNU GENERAL PUBLIC LICENSE)
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*)
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header {*
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  \title{Ring and field structures}
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  \author{Gertrud Bauer, L. C. Paulson and Markus Wenzel}
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*}
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theory Ring_and_Field = Inductive:
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subsection {* Abstract algebraic structures *}
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subsection {* Types Classes @{text plus_ac0} and @{text times_ac1} *}
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axclass plus_ac0 \<subseteq> plus, zero
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  commute:     "x + y = y + x"
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  assoc:       "(x + y) + z = x + (y + z)"
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  zero [simp]: "0 + x = x"
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lemma plus_ac0_left_commute: "x + (y+z) = y + ((x+z)::'a::plus_ac0)"
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apply (rule plus_ac0.commute [THEN trans])
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apply (rule plus_ac0.assoc [THEN trans])
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apply (rule plus_ac0.commute [THEN arg_cong])
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done
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lemma plus_ac0_zero_right [simp]: "x + 0 = (x ::'a::plus_ac0)"
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apply (rule plus_ac0.commute [THEN trans])
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apply (rule plus_ac0.zero)
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done
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lemmas plus_ac0 = plus_ac0.assoc plus_ac0.commute plus_ac0_left_commute
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                  plus_ac0.zero plus_ac0_zero_right
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axclass times_ac1 \<subseteq> times, one
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  commute:     "x * y = y * x"
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  assoc:       "(x * y) * z = x * (y * z)"
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  one [simp]:  "1 * x = x"
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theorem times_ac1_left_commute: "(x::'a::times_ac1) * ((y::'a) * z) =
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  y * (x * z)"
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proof -
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  have "(x::'a::times_ac1) * (y * z) = (x * y) * z"
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    by (rule times_ac1.assoc [THEN sym])
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  also have "x * y = y * x"
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    by (rule times_ac1.commute)
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  also have "(y * x) * z = y * (x * z)"
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    by (rule times_ac1.assoc)
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  finally show ?thesis .
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qed
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theorem times_ac1_one_right [simp]: "(x::'a::times_ac1) * 1 = x"
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proof -
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  have "x * 1 = 1 * x"
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    by (rule times_ac1.commute)
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  also have "... = x"
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    by (rule times_ac1.one)
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  finally show ?thesis .
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qed
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theorems times_ac1 = times_ac1.assoc times_ac1.commute times_ac1_left_commute
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  times_ac1.one times_ac1_one_right
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text{*This class is the same as @{text plus_ac0}, while using the same axiom
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names as the other axclasses.*}
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axclass abelian_semigroup \<subseteq> zero, plus
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  add_assoc: "(a + b) + c = a + (b + c)"
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  add_commute: "a + b = b + a"
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  add_0 [simp]: "0 + a = a"
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text{*This class underlies both @{text semiring} and @{text ring}*}
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axclass almost_semiring \<subseteq> abelian_semigroup, one, times
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  mult_assoc: "(a * b) * c = a * (b * c)"
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  mult_commute: "a * b = b * a"
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  mult_1 [simp]: "1 * a = a"
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  left_distrib: "(a + b) * c = a * c + b * c"
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  zero_neq_one [simp]: "0 \<noteq> 1"
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axclass semiring \<subseteq> almost_semiring
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  add_left_imp_eq: "a + b = a + c ==> b=c"
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    --{*This axiom is needed for semirings only: for rings, etc., it is
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        redundant. Including it allows many more of the following results
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        to be proved for semirings too.*}
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instance abelian_semigroup \<subseteq> plus_ac0
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proof
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  fix x y z :: 'a
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  show "x + y = y + x" by (rule add_commute)
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  show "x + y + z = x + (y + z)" by (rule add_assoc)
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  show "0+x = x" by (rule add_0)
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qed
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instance almost_semiring \<subseteq> times_ac1
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proof
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  fix x y z :: 'a
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  show "x * y = y * x" by (rule mult_commute)
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  show "x * y * z = x * (y * z)" by (rule mult_assoc)
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  show "1*x = x" by simp
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qed
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axclass abelian_group \<subseteq> abelian_semigroup, minus
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   left_minus [simp]: "-a + a = 0"
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   diff_minus: "a - b = a + -b"
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axclass ring \<subseteq> almost_semiring, abelian_group
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text{*Proving axiom @{text add_left_imp_eq} makes all @{text semiring}
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      theorems available to members of @{term ring} *}
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instance ring \<subseteq> semiring
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proof
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  fix a b c :: 'a
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  assume "a + b = a + c"
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  hence  "-a + a + b = -a + a + c" by (simp only: add_assoc)
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  thus "b = c" by simp
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qed
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text{*This class underlies @{text ordered_semiring} and @{text ordered_ring}*}
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axclass almost_ordered_semiring \<subseteq> semiring, linorder
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  add_left_mono: "a \<le> b ==> c + a \<le> c + b"
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  mult_strict_left_mono: "a < b ==> 0 < c ==> c * a < c * b"
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axclass ordered_semiring \<subseteq> almost_ordered_semiring
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  zero_less_one [simp]: "0 < 1" --{*This too is needed for semirings only.*}
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axclass ordered_ring \<subseteq> almost_ordered_semiring, ring
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  abs_if: "\<bar>a\<bar> = (if a < 0 then -a else a)"
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axclass field \<subseteq> ring, inverse
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  left_inverse [simp]: "a \<noteq> 0 ==> inverse a * a = 1"
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  divide_inverse:      "a / b = a * inverse b"
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axclass ordered_field \<subseteq> ordered_ring, field
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axclass division_by_zero \<subseteq> zero, inverse
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  inverse_zero [simp]: "inverse 0 = 0"
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subsection {* Derived Rules for Addition *}
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lemma add_0_right [simp]: "a + 0 = (a::'a::plus_ac0)"
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proof -
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  have "a + 0 = 0 + a" by (rule plus_ac0.commute)
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parents:
diff changeset
   149
  also have "... = a" by simp
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   150
  finally show ?thesis .
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   151
qed
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   152
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   153
lemma add_left_commute: "a + (b + c) = b + (a + (c::'a::plus_ac0))"
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   154
  by (rule mk_left_commute [of "op +", OF plus_ac0.assoc plus_ac0.commute])
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   155
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   156
theorems add_ac = add_assoc add_commute add_left_commute
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   157
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   158
lemma right_minus [simp]: "a + -(a::'a::abelian_group) = 0"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   159
proof -
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   160
  have "a + -a = -a + a" by (simp add: add_ac)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   161
  also have "... = 0" by simp
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   162
  finally show ?thesis .
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   163
qed
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   164
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   165
lemma right_minus_eq: "(a - b = 0) = (a = (b::'a::abelian_group))"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   166
proof
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   167
  have "a = a - b + b" by (simp add: diff_minus add_ac)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   168
  also assume "a - b = 0"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   169
  finally show "a = b" by simp
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   170
next
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   171
  assume "a = b"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   172
  thus "a - b = 0" by (simp add: diff_minus)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   173
qed
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   174
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   175
lemma add_left_cancel [simp]:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   176
     "(a + b = a + c) = (b = (c::'a::semiring))"
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   177
by (blast dest: add_left_imp_eq) 
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   178
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   179
lemma add_right_cancel [simp]:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   180
     "(b + a = c + a) = (b = (c::'a::semiring))"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   181
  by (simp add: add_commute)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   182
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   183
lemma minus_minus [simp]: "- (- (a::'a::abelian_group)) = a" 
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   184
apply (rule right_minus_eq [THEN iffD1]) 
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   185
apply (simp add: diff_minus) 
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   186
done
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   187
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   188
lemma equals_zero_I: "a+b = 0 ==> -a = (b::'a::abelian_group)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   189
apply (rule right_minus_eq [THEN iffD1, symmetric])
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   190
apply (simp add: diff_minus add_commute) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   191
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   192
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   193
lemma minus_zero [simp]: "- 0 = (0::'a::abelian_group)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   194
by (simp add: equals_zero_I)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   195
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   196
lemma diff_self [simp]: "a - (a::'a::abelian_group) = 0"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   197
  by (simp add: diff_minus)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   198
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   199
lemma diff_0 [simp]: "(0::'a::abelian_group) - a = -a"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   200
by (simp add: diff_minus)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   201
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   202
lemma diff_0_right [simp]: "a - (0::'a::abelian_group) = a" 
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   203
by (simp add: diff_minus)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   204
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   205
lemma diff_minus_eq_add [simp]: "a - - b = a + (b::'a::abelian_group)"
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   206
by (simp add: diff_minus)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   207
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   208
lemma neg_equal_iff_equal [simp]: "(-a = -b) = (a = (b::'a::abelian_group))" 
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   209
proof 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   210
  assume "- a = - b"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   211
  hence "- (- a) = - (- b)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   212
    by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   213
  thus "a=b" by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   214
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   215
  assume "a=b"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   216
  thus "-a = -b" by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   217
qed
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   218
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   219
lemma neg_equal_0_iff_equal [simp]: "(-a = 0) = (a = (0::'a::abelian_group))"
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   220
by (subst neg_equal_iff_equal [symmetric], simp)
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   221
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   222
lemma neg_0_equal_iff_equal [simp]: "(0 = -a) = (0 = (a::'a::abelian_group))"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   223
by (subst neg_equal_iff_equal [symmetric], simp)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   224
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   225
lemma add_minus_self [simp]: "a + b - b = (a::'a::abelian_group)"; 
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   226
  by (simp add: diff_minus add_assoc)
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   227
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   228
lemma add_minus_self_left [simp]:  "a + (b - a)  = (b::'a::abelian_group)";
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   229
by (simp add: diff_minus add_left_commute [of a]) 
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   230
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   231
lemma add_minus_self_right  [simp]:  "a + b - a  = (b::'a::abelian_group)";
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   232
by (simp add: diff_minus add_left_commute [of a] add_assoc) 
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   233
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   234
lemma minus_add_self [simp]: "a - b + b = (a::'a::abelian_group)"; 
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   235
by (simp add: diff_minus add_assoc) 
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   236
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   237
text{*The next two equations can make the simplifier loop!*}
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   238
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   239
lemma equation_minus_iff: "(a = - b) = (b = - (a::'a::abelian_group))"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   240
proof -
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   241
  have "(- (-a) = - b) = (- a = b)" by (rule neg_equal_iff_equal)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   242
  thus ?thesis by (simp add: eq_commute)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   243
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   244
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   245
lemma minus_equation_iff: "(- a = b) = (- (b::'a::abelian_group) = a)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   246
proof -
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   247
  have "(- a = - (-b)) = (a = -b)" by (rule neg_equal_iff_equal)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   248
  thus ?thesis by (simp add: eq_commute)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   249
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   250
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   251
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   252
subsection {* Derived rules for multiplication *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   253
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   254
lemma mult_1_right [simp]: "a * (1::'a::almost_semiring) = a"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   255
proof -
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   256
  have "a * 1 = 1 * a" by (simp add: mult_commute)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   257
  also have "... = a" by simp
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   258
  finally show ?thesis .
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   259
qed
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   260
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   261
lemma mult_left_commute: "a * (b * c) = b * (a * (c::'a::almost_semiring))"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   262
  by (rule mk_left_commute [of "op *", OF mult_assoc mult_commute])
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   263
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   264
theorems mult_ac = mult_assoc mult_commute mult_left_commute
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   265
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   266
lemma mult_zero_left [simp]: "0 * a = (0::'a::semiring)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   267
proof -
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   268
  have "0*a + 0*a = 0*a + 0"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   269
    by (simp add: left_distrib [symmetric])
14266
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
   270
  thus ?thesis by (simp only: add_left_cancel)
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   271
qed
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   272
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   273
lemma mult_zero_right [simp]: "a * 0 = (0::'a::semiring)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   274
  by (simp add: mult_commute)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   275
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   276
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   277
subsection {* Distribution rules *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   278
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   279
lemma right_distrib: "a * (b + c) = a * b + a * (c::'a::almost_semiring)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   280
proof -
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   281
  have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   282
  also have "... = b * a + c * a" by (simp only: left_distrib)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   283
  also have "... = a * b + a * c" by (simp add: mult_ac)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   284
  finally show ?thesis .
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   285
qed
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   286
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   287
theorems ring_distrib = right_distrib left_distrib
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   288
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   289
text{*For the @{text combine_numerals} simproc*}
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   290
lemma combine_common_factor:
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   291
     "a*e + (b*e + c) = (a+b)*e + (c::'a::almost_semiring)"
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   292
by (simp add: left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   293
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   294
lemma minus_add_distrib [simp]: "- (a + b) = -a + -(b::'a::abelian_group)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   295
apply (rule equals_zero_I)
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   296
apply (simp add: plus_ac0) 
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   297
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   298
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   299
lemma minus_mult_left: "- (a * b) = (-a) * (b::'a::ring)"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   300
apply (rule equals_zero_I)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   301
apply (simp add: left_distrib [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   302
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   303
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   304
lemma minus_mult_right: "- (a * b) = a * -(b::'a::ring)"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   305
apply (rule equals_zero_I)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   306
apply (simp add: right_distrib [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   307
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   308
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   309
lemma minus_mult_minus [simp]: "(- a) * (- b) = a * (b::'a::ring)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   310
  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   311
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   312
lemma minus_mult_commute: "(- a) * b = a * (- b::'a::ring)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   313
  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   314
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   315
lemma right_diff_distrib: "a * (b - c) = a * b - a * (c::'a::ring)"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   316
by (simp add: right_distrib diff_minus 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   317
              minus_mult_left [symmetric] minus_mult_right [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   318
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   319
lemma left_diff_distrib: "(a - b) * c = a * c - b * (c::'a::ring)"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   320
by (simp add: mult_commute [of _ c] right_diff_distrib) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   321
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   322
lemma minus_diff_eq [simp]: "- (a - b) = b - (a::'a::ring)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   323
by (simp add: diff_minus add_commute) 
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   324
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   325
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   326
subsection {* Ordering Rules for Addition *}
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   327
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   328
lemma add_right_mono: "a \<le> (b::'a::almost_ordered_semiring) ==> a + c \<le> b + c"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   329
by (simp add: add_commute [of _ c] add_left_mono)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   330
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   331
text {* non-strict, in both arguments *}
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   332
lemma add_mono:
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   333
     "[|a \<le> b;  c \<le> d|] ==> a + c \<le> b + (d::'a::almost_ordered_semiring)"
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   334
  apply (erule add_right_mono [THEN order_trans])
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   335
  apply (simp add: add_commute add_left_mono)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   336
  done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   337
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   338
lemma add_strict_left_mono:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   339
     "a < b ==> c + a < c + (b::'a::almost_ordered_semiring)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   340
 by (simp add: order_less_le add_left_mono) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   341
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   342
lemma add_strict_right_mono:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   343
     "a < b ==> a + c < b + (c::'a::almost_ordered_semiring)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   344
 by (simp add: add_commute [of _ c] add_strict_left_mono)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   345
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   346
text{*Strict monotonicity in both arguments*}
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   347
lemma add_strict_mono: "[|a<b; c<d|] ==> a + c < b + (d::'a::almost_ordered_semiring)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   348
apply (erule add_strict_right_mono [THEN order_less_trans])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   349
apply (erule add_strict_left_mono)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   350
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   351
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 14368
diff changeset
   352
lemma add_less_le_mono:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   353
     "[| a<b; c\<le>d |] ==> a + c < b + (d::'a::almost_ordered_semiring)"
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   354
apply (erule add_strict_right_mono [THEN order_less_le_trans])
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   355
apply (erule add_left_mono) 
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   356
done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   357
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   358
lemma add_le_less_mono:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   359
     "[| a\<le>b; c<d |] ==> a + c < b + (d::'a::almost_ordered_semiring)"
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   360
apply (erule add_right_mono [THEN order_le_less_trans])
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   361
apply (erule add_strict_left_mono) 
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   362
done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   363
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   364
lemma add_less_imp_less_left:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   365
      assumes less: "c + a < c + b"  shows "a < (b::'a::almost_ordered_semiring)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   366
proof (rule ccontr)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   367
  assume "~ a < b"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   368
  hence "b \<le> a" by (simp add: linorder_not_less)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   369
  hence "c+b \<le> c+a" by (rule add_left_mono)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   370
  with this and less show False 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   371
    by (simp add: linorder_not_less [symmetric])
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   372
qed
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   373
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   374
lemma add_less_imp_less_right:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   375
      "a + c < b + c ==> a < (b::'a::almost_ordered_semiring)"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   376
apply (rule add_less_imp_less_left [of c])
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   377
apply (simp add: add_commute)  
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   378
done
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   379
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   380
lemma add_less_cancel_left [simp]:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   381
    "(c+a < c+b) = (a < (b::'a::almost_ordered_semiring))"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   382
by (blast intro: add_less_imp_less_left add_strict_left_mono) 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   383
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   384
lemma add_less_cancel_right [simp]:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   385
    "(a+c < b+c) = (a < (b::'a::almost_ordered_semiring))"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   386
by (blast intro: add_less_imp_less_right add_strict_right_mono)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   387
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   388
lemma add_le_cancel_left [simp]:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   389
    "(c+a \<le> c+b) = (a \<le> (b::'a::almost_ordered_semiring))"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   390
by (simp add: linorder_not_less [symmetric]) 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   391
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   392
lemma add_le_cancel_right [simp]:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   393
    "(a+c \<le> b+c) = (a \<le> (b::'a::almost_ordered_semiring))"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   394
by (simp add: linorder_not_less [symmetric]) 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   395
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   396
lemma add_le_imp_le_left:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   397
      "c + a \<le> c + b ==> a \<le> (b::'a::almost_ordered_semiring)"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   398
by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   399
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   400
lemma add_le_imp_le_right:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   401
      "a + c \<le> b + c ==> a \<le> (b::'a::almost_ordered_semiring)"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   402
by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   403
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   404
lemma add_increasing: "[|0\<le>a; b\<le>c|] ==> b \<le> a + (c::'a::almost_ordered_semiring)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   405
by (insert add_mono [of 0 a b c], simp)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   406
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   407
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   408
subsection {* Ordering Rules for Unary Minus *}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   409
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   410
lemma le_imp_neg_le:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   411
      assumes "a \<le> (b::'a::ordered_ring)" shows "-b \<le> -a"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   412
proof -
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   413
  have "-a+a \<le> -a+b"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   414
    by (rule add_left_mono) 
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   415
  hence "0 \<le> -a+b"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   416
    by simp
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   417
  hence "0 + (-b) \<le> (-a + b) + (-b)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   418
    by (rule add_right_mono) 
14266
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
   419
  thus ?thesis
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   420
    by (simp add: add_assoc)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   421
qed
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   422
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   423
lemma neg_le_iff_le [simp]: "(-b \<le> -a) = (a \<le> (b::'a::ordered_ring))"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   424
proof 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   425
  assume "- b \<le> - a"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   426
  hence "- (- a) \<le> - (- b)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   427
    by (rule le_imp_neg_le)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   428
  thus "a\<le>b" by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   429
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   430
  assume "a\<le>b"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   431
  thus "-b \<le> -a" by (rule le_imp_neg_le)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   432
qed
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   433
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   434
lemma neg_le_0_iff_le [simp]: "(-a \<le> 0) = (0 \<le> (a::'a::ordered_ring))"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   435
by (subst neg_le_iff_le [symmetric], simp)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   436
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   437
lemma neg_0_le_iff_le [simp]: "(0 \<le> -a) = (a \<le> (0::'a::ordered_ring))"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   438
by (subst neg_le_iff_le [symmetric], simp)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   439
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   440
lemma neg_less_iff_less [simp]: "(-b < -a) = (a < (b::'a::ordered_ring))"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   441
by (force simp add: order_less_le) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   442
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   443
lemma neg_less_0_iff_less [simp]: "(-a < 0) = (0 < (a::'a::ordered_ring))"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   444
by (subst neg_less_iff_less [symmetric], simp)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   445
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   446
lemma neg_0_less_iff_less [simp]: "(0 < -a) = (a < (0::'a::ordered_ring))"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   447
by (subst neg_less_iff_less [symmetric], simp)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   448
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   449
text{*The next several equations can make the simplifier loop!*}
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   450
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   451
lemma less_minus_iff: "(a < - b) = (b < - (a::'a::ordered_ring))"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   452
proof -
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   453
  have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   454
  thus ?thesis by simp
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   455
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   456
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   457
lemma minus_less_iff: "(- a < b) = (- b < (a::'a::ordered_ring))"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   458
proof -
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   459
  have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   460
  thus ?thesis by simp
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   461
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   462
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   463
lemma le_minus_iff: "(a \<le> - b) = (b \<le> - (a::'a::ordered_ring))"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   464
apply (simp add: linorder_not_less [symmetric])
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   465
apply (rule minus_less_iff) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   466
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   467
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   468
lemma minus_le_iff: "(- a \<le> b) = (- b \<le> (a::'a::ordered_ring))"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   469
apply (simp add: linorder_not_less [symmetric])
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   470
apply (rule less_minus_iff) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   471
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   472
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   473
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   474
subsection{*Subtraction Laws*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   475
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   476
lemma add_diff_eq: "a + (b - c) = (a + b) - (c::'a::abelian_group)"
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   477
by (simp add: diff_minus plus_ac0)
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   478
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   479
lemma diff_add_eq: "(a - b) + c = (a + c) - (b::'a::abelian_group)"
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   480
by (simp add: diff_minus plus_ac0)
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   481
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   482
lemma diff_eq_eq: "(a-b = c) = (a = c + (b::'a::abelian_group))"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   483
by (auto simp add: diff_minus add_assoc)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   484
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   485
lemma eq_diff_eq: "(a = c-b) = (a + (b::'a::abelian_group) = c)"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   486
by (auto simp add: diff_minus add_assoc)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   487
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   488
lemma diff_diff_eq: "(a - b) - c = a - (b + (c::'a::abelian_group))"
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   489
by (simp add: diff_minus plus_ac0)
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   490
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   491
lemma diff_diff_eq2: "a - (b - c) = (a + c) - (b::'a::abelian_group)"
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   492
by (simp add: diff_minus plus_ac0)
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   493
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   494
text{*Further subtraction laws for ordered rings*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   495
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   496
lemma less_iff_diff_less_0: "(a < b) = (a - b < (0::'a::ordered_ring))"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   497
proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   498
  have  "(a < b) = (a + (- b) < b + (-b))"  
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   499
    by (simp only: add_less_cancel_right)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   500
  also have "... =  (a - b < 0)" by (simp add: diff_minus)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   501
  finally show ?thesis .
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   502
qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   503
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   504
lemma diff_less_eq: "(a-b < c) = (a < c + (b::'a::ordered_ring))"
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   505
apply (subst less_iff_diff_less_0)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   506
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   507
apply (simp add: diff_minus add_ac)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   508
done
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   509
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   510
lemma less_diff_eq: "(a < c-b) = (a + (b::'a::ordered_ring) < c)"
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   511
apply (subst less_iff_diff_less_0)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   512
apply (rule less_iff_diff_less_0 [of _ "c-b", THEN ssubst])
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   513
apply (simp add: diff_minus add_ac)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   514
done
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   515
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   516
lemma diff_le_eq: "(a-b \<le> c) = (a \<le> c + (b::'a::ordered_ring))"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   517
by (simp add: linorder_not_less [symmetric] less_diff_eq)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   518
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   519
lemma le_diff_eq: "(a \<le> c-b) = (a + (b::'a::ordered_ring) \<le> c)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   520
by (simp add: linorder_not_less [symmetric] diff_less_eq)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   521
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   522
text{*This list of rewrites simplifies (in)equalities by bringing subtractions
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   523
  to the top and then moving negative terms to the other side.
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   524
  Use with @{text add_ac}*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   525
lemmas compare_rls =
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   526
       diff_minus [symmetric]
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   527
       add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   528
       diff_less_eq less_diff_eq diff_le_eq le_diff_eq
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   529
       diff_eq_eq eq_diff_eq
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   530
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   531
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   532
subsection{*Lemmas for the @{text cancel_numerals} simproc*}
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   533
14504
7a3d80e276d4 new type class abelian_group
paulson
parents: 14475
diff changeset
   534
lemma eq_iff_diff_eq_0: "(a = b) = (a-b = (0::'a::abelian_group))"
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   535
by (simp add: compare_rls)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   536
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   537
lemma le_iff_diff_le_0: "(a \<le> b) = (a-b \<le> (0::'a::ordered_ring))"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   538
by (simp add: compare_rls)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   539
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   540
lemma eq_add_iff1:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   541
     "(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   542
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   543
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   544
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   545
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   546
lemma eq_add_iff2:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   547
     "(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   548
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   549
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   550
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   551
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   552
lemma less_add_iff1:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   553
     "(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::ordered_ring))"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   554
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   555
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   556
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   557
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   558
lemma less_add_iff2:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   559
     "(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::ordered_ring))"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   560
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   561
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   562
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   563
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   564
lemma le_add_iff1:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   565
     "(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::ordered_ring))"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   566
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   567
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   568
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   569
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   570
lemma le_add_iff2:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   571
     "(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::ordered_ring))"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   572
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   573
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   574
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   575
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   576
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   577
subsection {* Ordering Rules for Multiplication *}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   578
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   579
lemma mult_strict_right_mono:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   580
     "[|a < b; 0 < c|] ==> a * c < b * (c::'a::almost_ordered_semiring)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   581
by (simp add: mult_commute [of _ c] mult_strict_left_mono)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   582
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   583
lemma mult_left_mono:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   584
     "[|a \<le> b; 0 \<le> c|] ==> c * a \<le> c * (b::'a::almost_ordered_semiring)"
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   585
  apply (case_tac "c=0", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   586
  apply (force simp add: mult_strict_left_mono order_le_less) 
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   587
  done
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   588
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   589
lemma mult_right_mono:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   590
     "[|a \<le> b; 0 \<le> c|] ==> a*c \<le> b * (c::'a::almost_ordered_semiring)"
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   591
  by (simp add: mult_left_mono mult_commute [of _ c]) 
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   592
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   593
lemma mult_left_le_imp_le:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   594
     "[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::almost_ordered_semiring)"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   595
  by (force simp add: mult_strict_left_mono linorder_not_less [symmetric])
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   596
 
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   597
lemma mult_right_le_imp_le:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   598
     "[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::almost_ordered_semiring)"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   599
  by (force simp add: mult_strict_right_mono linorder_not_less [symmetric])
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   600
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   601
lemma mult_left_less_imp_less:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   602
     "[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::almost_ordered_semiring)"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   603
  by (force simp add: mult_left_mono linorder_not_le [symmetric])
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   604
 
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   605
lemma mult_right_less_imp_less:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   606
     "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::almost_ordered_semiring)"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   607
  by (force simp add: mult_right_mono linorder_not_le [symmetric])
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   608
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   609
lemma mult_strict_left_mono_neg:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   610
     "[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring)"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   611
apply (drule mult_strict_left_mono [of _ _ "-c"])
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   612
apply (simp_all add: minus_mult_left [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   613
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   614
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   615
lemma mult_strict_right_mono_neg:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   616
     "[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring)"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   617
apply (drule mult_strict_right_mono [of _ _ "-c"])
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   618
apply (simp_all add: minus_mult_right [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   619
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   620
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   621
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   622
subsection{* Products of Signs *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   623
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   624
lemma mult_pos: "[| (0::'a::almost_ordered_semiring) < a; 0 < b |] ==> 0 < a*b"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   625
by (drule mult_strict_left_mono [of 0 b], auto)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   626
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   627
lemma mult_pos_neg: "[| (0::'a::almost_ordered_semiring) < a; b < 0 |] ==> a*b < 0"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   628
by (drule mult_strict_left_mono [of b 0], auto)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   629
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   630
lemma mult_neg: "[| a < (0::'a::ordered_ring); b < 0 |] ==> 0 < a*b"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   631
by (drule mult_strict_right_mono_neg, auto)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   632
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   633
lemma zero_less_mult_pos:
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   634
     "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::almost_ordered_semiring)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   635
apply (case_tac "b\<le>0") 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   636
 apply (auto simp add: order_le_less linorder_not_less)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   637
apply (drule_tac mult_pos_neg [of a b]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   638
 apply (auto dest: order_less_not_sym)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   639
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   640
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   641
lemma zero_less_mult_iff:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   642
     "((0::'a::ordered_ring) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   643
apply (auto simp add: order_le_less linorder_not_less mult_pos mult_neg)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   644
apply (blast dest: zero_less_mult_pos) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   645
apply (simp add: mult_commute [of a b]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   646
apply (blast dest: zero_less_mult_pos) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   647
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   648
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   649
text{*A field has no "zero divisors", and this theorem holds without the
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   650
      assumption of an ordering.  See @{text field_mult_eq_0_iff} below.*}
14266
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
   651
lemma mult_eq_0_iff [simp]: "(a*b = (0::'a::ordered_ring)) = (a = 0 | b = 0)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   652
apply (case_tac "a < 0")
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   653
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   654
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   655
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   656
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   657
lemma zero_le_mult_iff:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   658
     "((0::'a::ordered_ring) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   659
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   660
                   zero_less_mult_iff)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   661
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   662
lemma mult_less_0_iff:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   663
     "(a*b < (0::'a::ordered_ring)) = (0 < a & b < 0 | a < 0 & 0 < b)"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   664
apply (insert zero_less_mult_iff [of "-a" b]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   665
apply (force simp add: minus_mult_left[symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   666
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   667
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   668
lemma mult_le_0_iff:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   669
     "(a*b \<le> (0::'a::ordered_ring)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   670
apply (insert zero_le_mult_iff [of "-a" b]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   671
apply (force simp add: minus_mult_left[symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   672
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   673
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   674
lemma zero_le_square: "(0::'a::ordered_ring) \<le> a*a"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   675
by (simp add: zero_le_mult_iff linorder_linear) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   676
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   677
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semiring}
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   678
      theorems available to members of @{term ordered_ring} *}
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   679
instance ordered_ring \<subseteq> ordered_semiring
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   680
proof
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   681
  have "(0::'a) \<le> 1*1" by (rule zero_le_square)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   682
  thus "(0::'a) < 1" by (simp add: order_le_less) 
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   683
qed
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   684
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   685
text{*All three types of comparision involving 0 and 1 are covered.*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   686
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   687
declare zero_neq_one [THEN not_sym, simp]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   688
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   689
lemma zero_le_one [simp]: "(0::'a::ordered_semiring) \<le> 1"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   690
  by (rule zero_less_one [THEN order_less_imp_le]) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   691
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   692
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semiring) \<le> 0"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   693
by (simp add: linorder_not_le zero_less_one) 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   694
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   695
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semiring) < 0"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   696
by (simp add: linorder_not_less zero_le_one) 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   697
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   698
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   699
subsection{*More Monotonicity*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   700
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   701
lemma mult_left_mono_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   702
     "[|b \<le> a; c \<le> 0|] ==> c * a \<le> c * (b::'a::ordered_ring)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   703
apply (drule mult_left_mono [of _ _ "-c"]) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   704
apply (simp_all add: minus_mult_left [symmetric]) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   705
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   706
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   707
lemma mult_right_mono_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   708
     "[|b \<le> a; c \<le> 0|] ==> a * c \<le> b * (c::'a::ordered_ring)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   709
  by (simp add: mult_left_mono_neg mult_commute [of _ c]) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   710
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   711
text{*Strict monotonicity in both arguments*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   712
lemma mult_strict_mono:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   713
     "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   714
apply (case_tac "c=0")
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   715
 apply (simp add: mult_pos) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   716
apply (erule mult_strict_right_mono [THEN order_less_trans])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   717
 apply (force simp add: order_le_less) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   718
apply (erule mult_strict_left_mono, assumption)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   719
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   720
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   721
text{*This weaker variant has more natural premises*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   722
lemma mult_strict_mono':
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   723
     "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   724
apply (rule mult_strict_mono)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   725
apply (blast intro: order_le_less_trans)+
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   726
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   727
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   728
lemma mult_mono:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   729
     "[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] 
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   730
      ==> a * c  \<le>  b * (d::'a::ordered_semiring)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   731
apply (erule mult_right_mono [THEN order_trans], assumption)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   732
apply (erule mult_left_mono, assumption)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   733
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   734
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   735
lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semiring)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   736
apply (insert mult_strict_mono [of 1 m 1 n]) 
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   737
apply (simp add:  order_less_trans [OF zero_less_one]) 
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   738
done
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   739
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   740
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   741
subsection{*Cancellation Laws for Relationships With a Common Factor*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   742
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   743
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   744
   also with the relations @{text "\<le>"} and equality.*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   745
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   746
lemma mult_less_cancel_right:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   747
    "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring)))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   748
apply (case_tac "c = 0")
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   749
apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   750
                      mult_strict_right_mono_neg)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   751
apply (auto simp add: linorder_not_less 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   752
                      linorder_not_le [symmetric, of "a*c"]
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   753
                      linorder_not_le [symmetric, of a])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   754
apply (erule_tac [!] notE)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   755
apply (auto simp add: order_less_imp_le mult_right_mono 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   756
                      mult_right_mono_neg)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   757
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   758
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   759
lemma mult_less_cancel_left:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   760
    "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring)))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   761
by (simp add: mult_commute [of c] mult_less_cancel_right)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   762
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   763
lemma mult_le_cancel_right:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   764
     "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring)))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   765
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   766
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   767
lemma mult_le_cancel_left:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   768
     "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring)))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   769
by (simp add: mult_commute [of c] mult_le_cancel_right)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   770
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   771
lemma mult_less_imp_less_left:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   772
      assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   773
      shows "a < (b::'a::ordered_semiring)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   774
proof (rule ccontr)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   775
  assume "~ a < b"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   776
  hence "b \<le> a" by (simp add: linorder_not_less)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   777
  hence "c*b \<le> c*a" by (rule mult_left_mono)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   778
  with this and less show False 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   779
    by (simp add: linorder_not_less [symmetric])
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   780
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   781
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   782
lemma mult_less_imp_less_right:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   783
    "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)"
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   784
  by (rule mult_less_imp_less_left, simp add: mult_commute)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   785
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   786
text{*Cancellation of equalities with a common factor*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   787
lemma mult_cancel_right [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   788
     "(a*c = b*c) = (c = (0::'a::ordered_ring) | a=b)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   789
apply (cut_tac linorder_less_linear [of 0 c])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   790
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   791
             simp add: linorder_neq_iff)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   792
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   793
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   794
text{*These cancellation theorems require an ordering. Versions are proved
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   795
      below that work for fields without an ordering.*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   796
lemma mult_cancel_left [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   797
     "(c*a = c*b) = (c = (0::'a::ordered_ring) | a=b)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   798
by (simp add: mult_commute [of c] mult_cancel_right)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   799
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   800
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   801
subsection {* Fields *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   802
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   803
lemma right_inverse [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   804
      assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   805
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   806
  have "a * inverse a = inverse a * a" by (simp add: mult_ac)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   807
  also have "... = 1" using not0 by simp
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   808
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   809
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   810
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   811
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   812
proof
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   813
  assume neq: "b \<noteq> 0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   814
  {
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   815
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   816
    also assume "a / b = 1"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   817
    finally show "a = b" by simp
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   818
  next
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   819
    assume "a = b"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   820
    with neq show "a / b = 1" by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   821
  }
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   822
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   823
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   824
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   825
by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   826
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   827
lemma divide_self [simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   828
  by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   829
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   830
lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   831
by (simp add: divide_inverse)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   832
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   833
lemma divide_zero_left [simp]: "0/a = (0::'a::field)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   834
by (simp add: divide_inverse)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   835
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   836
lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   837
by (simp add: divide_inverse)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   838
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   839
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c"
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   840
by (simp add: divide_inverse left_distrib) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   841
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   842
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   843
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   844
      of an ordering.*}
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   845
lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   846
proof cases
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   847
  assume "a=0" thus ?thesis by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   848
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   849
  assume anz [simp]: "a\<noteq>0"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   850
  { assume "a * b = 0"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   851
    hence "inverse a * (a * b) = 0" by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   852
    hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   853
  thus ?thesis by force
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   854
qed
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   855
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   856
text{*Cancellation of equalities with a common factor*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   857
lemma field_mult_cancel_right_lemma:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   858
      assumes cnz: "c \<noteq> (0::'a::field)"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   859
	  and eq:  "a*c = b*c"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   860
	 shows "a=b"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   861
proof -
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   862
  have "(a * c) * inverse c = (b * c) * inverse c"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   863
    by (simp add: eq)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   864
  thus "a=b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   865
    by (simp add: mult_assoc cnz)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   866
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   867
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   868
lemma field_mult_cancel_right [simp]:
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   869
     "(a*c = b*c) = (c = (0::'a::field) | a=b)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   870
proof cases
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   871
  assume "c=0" thus ?thesis by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   872
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   873
  assume "c\<noteq>0" 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   874
  thus ?thesis by (force dest: field_mult_cancel_right_lemma)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   875
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   876
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   877
lemma field_mult_cancel_left [simp]:
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   878
     "(c*a = c*b) = (c = (0::'a::field) | a=b)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   879
  by (simp add: mult_commute [of c] field_mult_cancel_right) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   880
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   881
lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   882
proof
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   883
  assume ianz: "inverse a = 0"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   884
  assume "a \<noteq> 0"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   885
  hence "1 = a * inverse a" by simp
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   886
  also have "... = 0" by (simp add: ianz)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   887
  finally have "1 = (0::'a::field)" .
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   888
  thus False by (simp add: eq_commute)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   889
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   890
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   891
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   892
subsection{*Basic Properties of @{term inverse}*}
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   893
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   894
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   895
apply (rule ccontr) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   896
apply (blast dest: nonzero_imp_inverse_nonzero) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   897
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   898
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   899
lemma inverse_nonzero_imp_nonzero:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   900
   "inverse a = 0 ==> a = (0::'a::field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   901
apply (rule ccontr) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   902
apply (blast dest: nonzero_imp_inverse_nonzero) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   903
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   904
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   905
lemma inverse_nonzero_iff_nonzero [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   906
   "(inverse a = 0) = (a = (0::'a::{field,division_by_zero}))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   907
by (force dest: inverse_nonzero_imp_nonzero) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   908
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   909
lemma nonzero_inverse_minus_eq:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   910
      assumes [simp]: "a\<noteq>0"  shows "inverse(-a) = -inverse(a::'a::field)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   911
proof -
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   912
  have "-a * inverse (- a) = -a * - inverse a"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   913
    by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   914
  thus ?thesis 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   915
    by (simp only: field_mult_cancel_left, simp)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   916
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   917
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   918
lemma inverse_minus_eq [simp]:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   919
   "inverse(-a) = -inverse(a::'a::{field,division_by_zero})";
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   920
proof cases
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   921
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   922
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   923
  assume "a\<noteq>0" 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   924
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   925
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   926
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   927
lemma nonzero_inverse_eq_imp_eq:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   928
      assumes inveq: "inverse a = inverse b"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   929
	  and anz:  "a \<noteq> 0"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   930
	  and bnz:  "b \<noteq> 0"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   931
	 shows "a = (b::'a::field)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   932
proof -
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   933
  have "a * inverse b = a * inverse a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   934
    by (simp add: inveq)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   935
  hence "(a * inverse b) * b = (a * inverse a) * b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   936
    by simp
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   937
  thus "a = b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   938
    by (simp add: mult_assoc anz bnz)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   939
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   940
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   941
lemma inverse_eq_imp_eq:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   942
     "inverse a = inverse b ==> a = (b::'a::{field,division_by_zero})"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   943
apply (case_tac "a=0 | b=0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   944
 apply (force dest!: inverse_zero_imp_zero
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   945
              simp add: eq_commute [of "0::'a"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   946
apply (force dest!: nonzero_inverse_eq_imp_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   947
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   948
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   949
lemma inverse_eq_iff_eq [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   950
     "(inverse a = inverse b) = (a = (b::'a::{field,division_by_zero}))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   951
by (force dest!: inverse_eq_imp_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   952
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   953
lemma nonzero_inverse_inverse_eq:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   954
      assumes [simp]: "a \<noteq> 0"  shows "inverse(inverse (a::'a::field)) = a"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   955
  proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   956
  have "(inverse (inverse a) * inverse a) * a = a" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   957
    by (simp add: nonzero_imp_inverse_nonzero)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   958
  thus ?thesis
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   959
    by (simp add: mult_assoc)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   960
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   961
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   962
lemma inverse_inverse_eq [simp]:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   963
     "inverse(inverse (a::'a::{field,division_by_zero})) = a"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   964
  proof cases
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   965
    assume "a=0" thus ?thesis by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   966
  next
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   967
    assume "a\<noteq>0" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   968
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   969
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   970
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   971
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   972
  proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   973
  have "inverse 1 * 1 = (1::'a::field)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   974
    by (rule left_inverse [OF zero_neq_one [symmetric]])
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   975
  thus ?thesis  by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   976
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   977
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   978
lemma nonzero_inverse_mult_distrib: 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   979
      assumes anz: "a \<noteq> 0"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   980
          and bnz: "b \<noteq> 0"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   981
      shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   982
  proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   983
  have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   984
    by (simp add: field_mult_eq_0_iff anz bnz)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   985
  hence "inverse(a*b) * a = inverse(b)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   986
    by (simp add: mult_assoc bnz)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   987
  hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   988
    by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   989
  thus ?thesis
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   990
    by (simp add: mult_assoc anz)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   991
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   992
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   993
text{*This version builds in division by zero while also re-orienting
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   994
      the right-hand side.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   995
lemma inverse_mult_distrib [simp]:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   996
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   997
  proof cases
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   998
    assume "a \<noteq> 0 & b \<noteq> 0" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   999
    thus ?thesis  by (simp add: nonzero_inverse_mult_distrib mult_commute)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1000
  next
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1001
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1002
    thus ?thesis  by force
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1003
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1004
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1005
text{*There is no slick version using division by zero.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1006
lemma inverse_add:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1007
     "[|a \<noteq> 0;  b \<noteq> 0|]
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1008
      ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1009
apply (simp add: left_distrib mult_assoc)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1010
apply (simp add: mult_commute [of "inverse a"]) 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1011
apply (simp add: mult_assoc [symmetric] add_commute)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1012
done
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1013
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1014
lemma inverse_divide [simp]:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1015
      "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1016
  by (simp add: divide_inverse mult_commute)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1017
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1018
lemma nonzero_mult_divide_cancel_left:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1019
  assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1020
    shows "(c*a)/(c*b) = a/(b::'a::field)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1021
proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1022
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1023
    by (simp add: field_mult_eq_0_iff divide_inverse 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1024
                  nonzero_inverse_mult_distrib)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1025
  also have "... =  a * inverse b * (inverse c * c)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1026
    by (simp only: mult_ac)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1027
  also have "... =  a * inverse b"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1028
    by simp
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1029
    finally show ?thesis 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1030
    by (simp add: divide_inverse)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1031
qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1032
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1033
lemma mult_divide_cancel_left:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1034
     "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1035
apply (case_tac "b = 0")
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1036
apply (simp_all add: nonzero_mult_divide_cancel_left)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1037
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1038
14321
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
  1039
lemma nonzero_mult_divide_cancel_right:
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
  1040
     "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
  1041
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
  1042
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
  1043
lemma mult_divide_cancel_right:
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
  1044
     "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
  1045
apply (case_tac "b = 0")
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
  1046
apply (simp_all add: nonzero_mult_divide_cancel_right)
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
  1047
done
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
  1048
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1049
(*For ExtractCommonTerm*)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1050
lemma mult_divide_cancel_eq_if:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1051
     "(c*a) / (c*b) = 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1052
      (if c=0 then 0 else a / (b::'a::{field,division_by_zero}))"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1053
  by (simp add: mult_divide_cancel_left)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1054
14284
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
  1055
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1056
  by (simp add: divide_inverse)
14284
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
  1057
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1058
lemma times_divide_eq_right [simp]: "a * (b/c) = (a*b) / (c::'a::field)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1059
by (simp add: divide_inverse mult_assoc)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1060
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1061
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1062
by (simp add: divide_inverse mult_ac)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1063
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1064
lemma divide_divide_eq_right [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1065
     "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1066
by (simp add: divide_inverse mult_ac)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1067
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1068
lemma divide_divide_eq_left [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1069
     "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1070
by (simp add: divide_inverse mult_assoc)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1071
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1072
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1073
subsection {* Division and Unary Minus *}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1074
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1075
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1076
by (simp add: divide_inverse minus_mult_left)
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1077
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1078
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1079
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1080
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1081
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1082
by (simp add: divide_inverse nonzero_inverse_minus_eq)
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1083
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1084
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1085
by (simp add: divide_inverse minus_mult_left [symmetric])
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1086
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1087
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1088
by (simp add: divide_inverse minus_mult_right [symmetric])
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1089
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1090
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1091
text{*The effect is to extract signs from divisions*}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1092
declare minus_divide_left  [symmetric, simp]
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1093
declare minus_divide_right [symmetric, simp]
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1094
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1095
text{*Also, extract signs from products*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1096
declare minus_mult_left [symmetric, simp]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1097
declare minus_mult_right [symmetric, simp]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1098
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1099
lemma minus_divide_divide [simp]:
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1100
     "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1101
apply (case_tac "b=0", simp) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1102
apply (simp add: nonzero_minus_divide_divide) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1103
done
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1104
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1105
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1106
by (simp add: diff_minus add_divide_distrib) 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1107
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1108
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1109
subsection {* Ordered Fields *}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1110
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1111
lemma positive_imp_inverse_positive: 
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1112
      assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1113
  proof -
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1114
  have "0 < a * inverse a" 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1115
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1116
  thus "0 < inverse a" 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1117
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1118
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1119
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1120
lemma negative_imp_inverse_negative:
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1121
     "a < 0 ==> inverse a < (0::'a::ordered_field)"
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1122
  by (insert positive_imp_inverse_positive [of "-a"], 
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1123
      simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1124
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1125
lemma inverse_le_imp_le:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1126
      assumes invle: "inverse a \<le> inverse b"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1127
	  and apos:  "0 < a"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1128
	 shows "b \<le> (a::'a::ordered_field)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1129
  proof (rule classical)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1130
  assume "~ b \<le> a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1131
  hence "a < b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1132
    by (simp add: linorder_not_le)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1133
  hence bpos: "0 < b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1134
    by (blast intro: apos order_less_trans)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1135
  hence "a * inverse a \<le> a * inverse b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1136
    by (simp add: apos invle order_less_imp_le mult_left_mono)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1137
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1138
    by (simp add: bpos order_less_imp_le mult_right_mono)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1139
  thus "b \<le> a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1140
    by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1141
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1142
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1143
lemma inverse_positive_imp_positive:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1144
      assumes inv_gt_0: "0 < inverse a"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1145
          and [simp]:   "a \<noteq> 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1146
        shows "0 < (a::'a::ordered_field)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1147
  proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1148
  have "0 < inverse (inverse a)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1149
    by (rule positive_imp_inverse_positive)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1150
  thus "0 < a"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1151
    by (simp add: nonzero_inverse_inverse_eq)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1152
  qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1153
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1154
lemma inverse_positive_iff_positive [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1155
      "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1156
apply (case_tac "a = 0", simp)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1157
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
  1158
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1159
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1160
lemma inverse_negative_imp_negative:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1161
      assumes inv_less_0: "inverse a < 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1162
          and [simp]:   "a \<noteq> 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1163
        shows "a < (0::'a::ordered_field)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1164
  proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1165
  have "inverse (inverse a) < 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1166
    by (rule negative_imp_inverse_negative)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1167
  thus "a < 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1168
    by (simp add: nonzero_inverse_inverse_eq)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1169
  qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1170
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1171
lemma inverse_negative_iff_negative [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1172
      "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1173
apply (case_tac "a = 0", simp)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1174
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
  1175
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1176
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1177
lemma inverse_nonnegative_iff_nonnegative [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1178
      "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1179
by (simp add: linorder_not_less [symmetric])
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1180
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1181
lemma inverse_nonpositive_iff_nonpositive [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1182
      "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1183
by (simp add: linorder_not_less [symmetric])
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1184
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1185
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1186
subsection{*Anti-Monotonicity of @{term inverse}*}
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1187
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1188
lemma less_imp_inverse_less:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1189
      assumes less: "a < b"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1190
	  and apos:  "0 < a"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1191
	shows "inverse b < inverse (a::'a::ordered_field)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1192
  proof (rule ccontr)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1193
  assume "~ inverse b < inverse a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1194
  hence "inverse a \<le> inverse b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1195
    by (simp add: linorder_not_less)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1196
  hence "~ (a < b)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1197
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1198
  thus False
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1199
    by (rule notE [OF _ less])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1200
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1201
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1202
lemma inverse_less_imp_less:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1203
   "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1204
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1205
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1206
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1207
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1208
text{*Both premises are essential. Consider -1 and 1.*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1209
lemma inverse_less_iff_less [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1210
     "[|0 < a; 0 < b|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1211
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1212
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1213
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1214
lemma le_imp_inverse_le:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1215
   "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1216
  by (force simp add: order_le_less less_imp_inverse_less)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1217
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1218
lemma inverse_le_iff_le [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1219
     "[|0 < a; 0 < b|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1220
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1221
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1222
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1223
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1224
text{*These results refer to both operands being negative.  The opposite-sign
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1225
case is trivial, since inverse preserves signs.*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1226
lemma inverse_le_imp_le_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1227
   "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1228
  apply (rule classical) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1229
  apply (subgoal_tac "a < 0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1230
   prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1231
  apply (insert inverse_le_imp_le [of "-b" "-a"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1232
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1233
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1234
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1235
lemma less_imp_inverse_less_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1236
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1237
  apply (subgoal_tac "a < 0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1238
   prefer 2 apply (blast intro: order_less_trans) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1239
  apply (insert less_imp_inverse_less [of "-b" "-a"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1240
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1241
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1242
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1243
lemma inverse_less_imp_less_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1244
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1245
  apply (rule classical) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1246
  apply (subgoal_tac "a < 0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1247
   prefer 2
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1248
   apply (force simp add: linorder_not_less intro: order_le_less_trans) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1249
  apply (insert inverse_less_imp_less [of "-b" "-a"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1250
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1251
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1252
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1253
lemma inverse_less_iff_less_neg [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1254
     "[|a < 0; b < 0|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1255
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1256
  apply (insert inverse_less_iff_less [of "-b" "-a"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1257
  apply (simp del: inverse_less_iff_less 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1258
	      add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1259
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1260
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1261
lemma le_imp_inverse_le_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1262
   "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1263
  by (force simp add: order_le_less less_imp_inverse_less_neg)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1264
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1265
lemma inverse_le_iff_le_neg [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1266
     "[|a < 0; b < 0|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1267
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1268
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
  1269
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1270
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1271
subsection{*Inverses and the Number One*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1272
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1273
lemma one_less_inverse_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1274
    "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1275
  assume "0 < x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1276
    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
  1277
    show ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1278
next
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1279
  assume notless: "~ (0 < x)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1280
  have "~ (1 < inverse x)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1281
  proof
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1282
    assume "1 < inverse x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1283
    also with notless have "... \<le> 0" by (simp add: linorder_not_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1284
    also have "... < 1" by (rule zero_less_one) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1285
    finally show False by auto
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1286
  qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1287
  with notless show ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1288
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1289
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1290
lemma inverse_eq_1_iff [simp]:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: