src/HOL/Ring_and_Field.thy
author paulson
Thu, 05 Feb 2004 10:45:28 +0100
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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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axclass semiring \<subseteq> zero, one, plus, times
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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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  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. The drawback is that this redundant
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        axiom must be proved for instances of rings.*}
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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 ring \<subseteq> semiring, 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 ordered_semiring \<subseteq> semiring, linorder
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  zero_less_one: "0 < 1" --{*This axiom too is needed for semirings only.*}
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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_ring \<subseteq> 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:      "b \<noteq> 0 ==> 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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  divide_zero [simp]: "a / 0 = 0"
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subsection {* Derived Rules for Addition *}
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lemma add_0_right [simp]: "a + 0 = (a::'a::semiring)"
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proof -
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  have "a + 0 = 0 + a" by (simp only: add_commute)
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  also have "... = a" by simp
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  finally show ?thesis .
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qed
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lemma add_left_commute: "a + (b + c) = b + (a + (c::'a::semiring))"
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  by (rule mk_left_commute [of "op +", OF add_assoc add_commute])
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theorems add_ac = add_assoc add_commute add_left_commute
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lemma right_minus [simp]: "a + -(a::'a::ring) = 0"
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proof -
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  have "a + -a = -a + a" by (simp add: add_ac)
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  also have "... = 0" by simp
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  finally show ?thesis .
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qed
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lemma right_minus_eq: "(a - b = 0) = (a = (b::'a::ring))"
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proof
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  have "a = a - b + b" by (simp add: diff_minus add_ac)
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  also assume "a - b = 0"
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  finally show "a = b" by simp
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next
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  assume "a = b"
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  thus "a - b = 0" by (simp add: diff_minus)
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qed
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lemma add_left_cancel [simp]:
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     "(a + b = a + c) = (b = (c::'a::semiring))"
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by (blast dest: add_left_imp_eq) 
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lemma add_right_cancel [simp]:
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     "(b + a = c + a) = (b = (c::'a::semiring))"
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  by (simp add: add_commute)
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lemma minus_minus [simp]: "- (- (a::'a::ring)) = a"
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proof (rule add_left_cancel [of "-a", THEN iffD1])
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  show "(-a + -(-a) = -a + a)"
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  by simp
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qed
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lemma equals_zero_I: "a+b = 0 ==> -a = (b::'a::ring)"
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apply (rule right_minus_eq [THEN iffD1, symmetric])
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apply (simp add: diff_minus add_commute) 
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done
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lemma minus_zero [simp]: "- 0 = (0::'a::ring)"
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by (simp add: equals_zero_I)
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lemma diff_self [simp]: "a - (a::'a::ring) = 0"
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  by (simp add: diff_minus)
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lemma diff_0 [simp]: "(0::'a::ring) - a = -a"
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by (simp add: diff_minus)
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lemma diff_0_right [simp]: "a - (0::'a::ring) = a" 
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by (simp add: diff_minus)
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lemma diff_minus_eq_add [simp]: "a - - b = a + (b::'a::ring)"
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by (simp add: diff_minus)
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lemma neg_equal_iff_equal [simp]: "(-a = -b) = (a = (b::'a::ring))" 
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proof 
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  assume "- a = - b"
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  hence "- (- a) = - (- b)"
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    by simp
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  thus "a=b" by simp
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next
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  assume "a=b"
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  thus "-a = -b" by simp
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qed
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lemma neg_equal_0_iff_equal [simp]: "(-a = 0) = (a = (0::'a::ring))"
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by (subst neg_equal_iff_equal [symmetric], simp)
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lemma neg_0_equal_iff_equal [simp]: "(0 = -a) = (0 = (a::'a::ring))"
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paulson
parents:
diff changeset
   137
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
   138
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   139
text{*The next two equations can make the simplifier loop!*}
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   140
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   141
lemma equation_minus_iff: "(a = - b) = (b = - (a::'a::ring))"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   142
proof -
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   143
  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
   144
  thus ?thesis by (simp add: eq_commute)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   145
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   146
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   147
lemma minus_equation_iff: "(- a = b) = (- (b::'a::ring) = a)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   148
proof -
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   149
  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
   150
  thus ?thesis by (simp add: eq_commute)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   151
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   152
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   153
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   154
subsection {* Derived rules for multiplication *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   155
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   156
lemma mult_1_right [simp]: "a * (1::'a::semiring) = a"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   157
proof -
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   158
  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
   159
  also have "... = a" by simp
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   160
  finally show ?thesis .
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   161
qed
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   162
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   163
lemma mult_left_commute: "a * (b * c) = b * (a * (c::'a::semiring))"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   164
  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
   165
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   166
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
   167
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   168
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
   169
proof -
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   170
  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
   171
    by (simp add: left_distrib [symmetric])
14266
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
   172
  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
   173
qed
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   174
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   175
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
   176
  by (simp add: mult_commute)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   177
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
subsection {* Distribution rules *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   180
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   181
lemma right_distrib: "a * (b + c) = a * b + a * (c::'a::semiring)"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   182
proof -
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   183
  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
   184
  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
   185
  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
   186
  finally show ?thesis .
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   187
qed
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   188
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   189
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
   190
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   191
text{*For the @{text combine_numerals} simproc*}
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   192
lemma combine_common_factor: "a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   193
by (simp add: left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   194
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   195
lemma minus_add_distrib [simp]: "- (a + b) = -a + -(b::'a::ring)"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   196
apply (rule equals_zero_I)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   197
apply (simp add: add_ac) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   198
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   199
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   200
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
   201
apply (rule equals_zero_I)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   202
apply (simp add: left_distrib [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   203
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   204
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   205
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
   206
apply (rule equals_zero_I)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   207
apply (simp add: right_distrib [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   208
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   209
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   210
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
   211
  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
   212
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   213
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
   214
  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
   215
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   216
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
   217
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
   218
              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
   219
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   220
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
   221
by (simp add: mult_commute [of _ c] right_diff_distrib) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   222
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   223
lemma minus_diff_eq [simp]: "- (a - b) = b - (a::'a::ring)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   224
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
   225
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   226
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   227
subsection {* Ordering Rules for Addition *}
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   228
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   229
lemma add_right_mono: "a \<le> (b::'a::ordered_semiring) ==> a + c \<le> b + c"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   230
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
   231
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   232
text {* non-strict, in both arguments *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   233
lemma add_mono: "[|a \<le> b;  c \<le> d|] ==> a + c \<le> b + (d::'a::ordered_semiring)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   234
  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
   235
  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
   236
  done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   237
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   238
lemma add_strict_left_mono:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   239
     "a < b ==> c + a < c + (b::'a::ordered_semiring)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   240
 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
   241
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   242
lemma add_strict_right_mono:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   243
     "a < b ==> a + c < b + (c::'a::ordered_semiring)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   244
 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
   245
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   246
text{*Strict monotonicity in both arguments*}
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   247
lemma add_strict_mono: "[|a<b; c<d|] ==> 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
   248
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
   249
apply (erule add_strict_left_mono)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   250
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   251
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 14368
diff changeset
   252
lemma add_less_le_mono:
b0064703967b simplifications in the hyperreals
paulson
parents: 14368
diff changeset
   253
     "[| a<b; c\<le>d |] ==> a + c < b + (d::'a::ordered_semiring)"
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   254
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
   255
apply (erule add_left_mono) 
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   256
done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   257
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   258
lemma add_le_less_mono:
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   259
     "[| a\<le>b; c<d |] ==> a + c < b + (d::'a::ordered_semiring)"
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   260
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
   261
apply (erule add_strict_left_mono) 
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   262
done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   263
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   264
lemma add_less_imp_less_left:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   265
      assumes less: "c + a < c + b"  shows "a < (b::'a::ordered_semiring)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   266
proof (rule ccontr)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   267
  assume "~ a < b"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   268
  hence "b \<le> a" by (simp add: linorder_not_less)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   269
  hence "c+b \<le> c+a" by (rule add_left_mono)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   270
  with this and less show False 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   271
    by (simp add: linorder_not_less [symmetric])
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   272
qed
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   273
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   274
lemma add_less_imp_less_right:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   275
      "a + c < b + c ==> a < (b::'a::ordered_semiring)"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   276
apply (rule add_less_imp_less_left [of c])
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   277
apply (simp add: add_commute)  
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   278
done
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   279
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   280
lemma add_less_cancel_left [simp]:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   281
    "(c+a < c+b) = (a < (b::'a::ordered_semiring))"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   282
by (blast intro: add_less_imp_less_left add_strict_left_mono) 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   283
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   284
lemma add_less_cancel_right [simp]:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   285
    "(a+c < b+c) = (a < (b::'a::ordered_semiring))"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   286
by (blast intro: add_less_imp_less_right add_strict_right_mono)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   287
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   288
lemma add_le_cancel_left [simp]:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   289
    "(c+a \<le> c+b) = (a \<le> (b::'a::ordered_semiring))"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   290
by (simp add: linorder_not_less [symmetric]) 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   291
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   292
lemma add_le_cancel_right [simp]:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   293
    "(a+c \<le> b+c) = (a \<le> (b::'a::ordered_semiring))"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   294
by (simp add: linorder_not_less [symmetric]) 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   295
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   296
lemma add_le_imp_le_left:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   297
      "c + a \<le> c + b ==> a \<le> (b::'a::ordered_semiring)"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   298
by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   299
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   300
lemma add_le_imp_le_right:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   301
      "a + c \<le> b + c ==> a \<le> (b::'a::ordered_semiring)"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   302
by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   303
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   304
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   305
subsection {* Ordering Rules for Unary Minus *}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   306
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   307
lemma le_imp_neg_le:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   308
      assumes "a \<le> (b::'a::ordered_ring)" shows "-b \<le> -a"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   309
proof -
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   310
  have "-a+a \<le> -a+b"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   311
    by (rule add_left_mono) 
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   312
  hence "0 \<le> -a+b"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   313
    by simp
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   314
  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
   315
    by (rule add_right_mono) 
14266
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
   316
  thus ?thesis
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   317
    by (simp add: add_assoc)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   318
qed
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   319
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   320
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
   321
proof 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   322
  assume "- b \<le> - a"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   323
  hence "- (- a) \<le> - (- b)"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   324
    by (rule le_imp_neg_le)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   325
  thus "a\<le>b" by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   326
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   327
  assume "a\<le>b"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   328
  thus "-b \<le> -a" by (rule le_imp_neg_le)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   329
qed
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   330
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   331
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
   332
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
   333
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   334
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
   335
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
   336
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   337
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
   338
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
   339
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   340
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
   341
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
   342
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   343
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
   344
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
   345
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   346
text{*The next several equations can make the simplifier loop!*}
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   347
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   348
lemma less_minus_iff: "(a < - b) = (b < - (a::'a::ordered_ring))"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   349
proof -
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   350
  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
   351
  thus ?thesis by simp
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   352
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   353
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   354
lemma minus_less_iff: "(- a < b) = (- b < (a::'a::ordered_ring))"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   355
proof -
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   356
  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
   357
  thus ?thesis by simp
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   358
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   359
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   360
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
   361
apply (simp add: linorder_not_less [symmetric])
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   362
apply (rule minus_less_iff) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   363
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   364
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   365
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
   366
apply (simp add: linorder_not_less [symmetric])
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   367
apply (rule less_minus_iff) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   368
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   369
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   370
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   371
subsection{*Subtraction Laws*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   372
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   373
lemma add_diff_eq: "a + (b - c) = (a + b) - (c::'a::ring)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   374
by (simp add: diff_minus add_ac)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   375
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   376
lemma diff_add_eq: "(a - b) + c = (a + c) - (b::'a::ring)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   377
by (simp add: diff_minus add_ac)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   378
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   379
lemma diff_eq_eq: "(a-b = c) = (a = c + (b::'a::ring))"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   380
by (auto simp add: diff_minus add_assoc)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   381
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   382
lemma eq_diff_eq: "(a = c-b) = (a + (b::'a::ring) = c)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   383
by (auto simp add: diff_minus add_assoc)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   384
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   385
lemma diff_diff_eq: "(a - b) - c = a - (b + (c::'a::ring))"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   386
by (simp add: diff_minus add_ac)
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 diff_diff_eq2: "a - (b - c) = (a + c) - (b::'a::ring)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   389
by (simp add: diff_minus add_ac)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   390
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   391
text{*Further subtraction laws for ordered rings*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   392
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   393
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
   394
proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   395
  have  "(a < b) = (a + (- b) < b + (-b))"  
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   396
    by (simp only: add_less_cancel_right)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   397
  also have "... =  (a - b < 0)" by (simp add: diff_minus)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   398
  finally show ?thesis .
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   399
qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   400
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   401
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
   402
apply (subst less_iff_diff_less_0)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   403
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   404
apply (simp add: diff_minus add_ac)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   405
done
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   406
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   407
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
   408
apply (subst less_iff_diff_less_0)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   409
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
   410
apply (simp add: diff_minus add_ac)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   411
done
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   412
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   413
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
   414
by (simp add: linorder_not_less [symmetric] less_diff_eq)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   415
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   416
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
   417
by (simp add: linorder_not_less [symmetric] diff_less_eq)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   418
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   419
text{*This list of rewrites simplifies (in)equalities by bringing subtractions
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   420
  to the top and then moving negative terms to the other side.
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   421
  Use with @{text add_ac}*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   422
lemmas compare_rls =
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   423
       diff_minus [symmetric]
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   424
       add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   425
       diff_less_eq less_diff_eq diff_le_eq le_diff_eq
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   426
       diff_eq_eq eq_diff_eq
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   427
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   428
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   429
subsection{*Lemmas for the @{text cancel_numerals} simproc*}
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   430
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   431
lemma eq_iff_diff_eq_0: "(a = b) = (a-b = (0::'a::ring))"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   432
by (simp add: compare_rls)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   433
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   434
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
   435
by (simp add: compare_rls)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   436
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   437
lemma eq_add_iff1:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   438
     "(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
   439
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   440
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   441
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   442
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   443
lemma eq_add_iff2:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   444
     "(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
   445
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   446
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   447
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   448
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   449
lemma less_add_iff1:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   450
     "(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
   451
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   452
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   453
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   454
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   455
lemma less_add_iff2:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   456
     "(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
   457
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   458
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   459
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   460
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   461
lemma le_add_iff1:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   462
     "(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
   463
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   464
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   465
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   466
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   467
lemma le_add_iff2:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   468
     "(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
   469
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   470
apply (simp add: compare_rls minus_mult_left [symmetric]) 
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
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   473
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   474
subsection {* Ordering Rules for Multiplication *}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   475
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   476
lemma mult_strict_right_mono:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   477
     "[|a < b; 0 < c|] ==> a * c < b * (c::'a::ordered_semiring)"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   478
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
   479
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   480
lemma mult_left_mono:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   481
     "[|a \<le> b; 0 \<le> c|] ==> c * a \<le> c * (b::'a::ordered_semiring)"
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   482
  apply (case_tac "c=0", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   483
  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
   484
  done
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   485
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   486
lemma mult_right_mono:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   487
     "[|a \<le> b; 0 \<le> c|] ==> a*c \<le> b * (c::'a::ordered_semiring)"
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   488
  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
   489
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   490
lemma mult_left_le_imp_le:
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   491
     "[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring)"
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   492
  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
   493
 
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   494
lemma mult_right_le_imp_le:
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   495
     "[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring)"
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   496
  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
   497
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   498
lemma mult_left_less_imp_less:
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   499
     "[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)"
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   500
  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
   501
 
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   502
lemma mult_right_less_imp_less:
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   503
     "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)"
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   504
  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
   505
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   506
lemma mult_strict_left_mono_neg:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   507
     "[|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
   508
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
   509
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
   510
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   511
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   512
lemma mult_strict_right_mono_neg:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   513
     "[|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
   514
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
   515
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
   516
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   517
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   518
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   519
subsection{* Products of Signs *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   520
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   521
lemma mult_pos: "[| (0::'a::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
   522
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
   523
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   524
lemma mult_pos_neg: "[| (0::'a::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
   525
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
   526
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   527
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
   528
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
   529
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   530
lemma zero_less_mult_pos:
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   531
     "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   532
apply (case_tac "b\<le>0") 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   533
 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
   534
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
   535
 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
   536
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   537
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   538
lemma zero_less_mult_iff:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   539
     "((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
   540
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
   541
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
   542
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
   543
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
   544
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   545
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   546
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
   547
      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
   548
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
   549
apply (case_tac "a < 0")
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   550
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
   551
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
   552
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   553
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   554
lemma zero_le_mult_iff:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   555
     "((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
   556
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
   557
                   zero_less_mult_iff)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   558
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   559
lemma mult_less_0_iff:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   560
     "(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
   561
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
   562
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
   563
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   564
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   565
lemma mult_le_0_iff:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   566
     "(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
   567
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
   568
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
   569
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   570
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   571
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
   572
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
   573
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   574
lemma zero_le_one: "(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
   575
  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
   576
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   577
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   578
subsection{*More Monotonicity*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   579
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   580
lemma mult_left_mono_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   581
     "[|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
   582
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
   583
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
   584
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   585
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   586
lemma mult_right_mono_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   587
     "[|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
   588
  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
   589
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   590
text{*Strict monotonicity in both arguments*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   591
lemma mult_strict_mono:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   592
     "[|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
   593
apply (case_tac "c=0")
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   594
 apply (simp add: mult_pos) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   595
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
   596
 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
   597
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
   598
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   599
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   600
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
   601
lemma mult_strict_mono':
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   602
     "[| 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
   603
apply (rule mult_strict_mono)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   604
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
   605
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   606
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   607
lemma mult_mono:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   608
     "[|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
   609
      ==> 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
   610
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
   611
apply (erule mult_left_mono, assumption)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   612
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   613
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   614
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   615
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
   616
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   617
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
   618
   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
   619
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   620
lemma mult_less_cancel_right:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   621
    "(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
   622
apply (case_tac "c = 0")
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   623
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
   624
                      mult_strict_right_mono_neg)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   625
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
   626
                      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
   627
                      linorder_not_le [symmetric, of a])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   628
apply (erule_tac [!] notE)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   629
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
   630
                      mult_right_mono_neg)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   631
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   632
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   633
lemma mult_less_cancel_left:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   634
    "(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
   635
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
   636
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   637
lemma mult_le_cancel_right:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   638
     "(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
   639
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
   640
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   641
lemma mult_le_cancel_left:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   642
     "(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
   643
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
   644
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   645
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
   646
      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
   647
      shows "a < (b::'a::ordered_semiring)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   648
proof (rule ccontr)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   649
  assume "~ a < b"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   650
  hence "b \<le> a" by (simp add: linorder_not_less)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   651
  hence "c*b \<le> c*a" by (rule mult_left_mono)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   652
  with this and less show False 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   653
    by (simp add: linorder_not_less [symmetric])
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   654
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   655
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   656
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
   657
    "[|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
   658
  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
   659
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   660
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
   661
lemma mult_cancel_right [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   662
     "(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
   663
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
   664
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
   665
             simp add: linorder_neq_iff)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   666
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   667
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   668
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
   669
      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
   670
lemma mult_cancel_left [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   671
     "(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
   672
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
   673
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   674
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   675
subsection {* Fields *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   676
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   677
lemma right_inverse [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   678
      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
   679
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   680
  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
   681
  also have "... = 1" using not0 by simp
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   682
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   683
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   684
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   685
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
   686
proof
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   687
  assume neq: "b \<noteq> 0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   688
  {
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   689
    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
   690
    also assume "a / b = 1"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   691
    finally show "a = b" by simp
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   692
  next
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   693
    assume "a = b"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   694
    with neq show "a / b = 1" by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   695
  }
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   696
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   697
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   698
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
   699
by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   700
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   701
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
   702
  by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   703
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   704
lemma divide_inverse_zero: "a/b = a * inverse(b::'a::{field,division_by_zero})"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   705
apply (case_tac "b = 0")
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   706
apply (simp_all add: divide_inverse)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   707
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   708
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   709
lemma divide_zero_left [simp]: "0/a = (0::'a::{field,division_by_zero})"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   710
by (simp add: divide_inverse_zero)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   711
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   712
lemma inverse_eq_divide: "inverse (a::'a::{field,division_by_zero}) = 1/a"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   713
by (simp add: divide_inverse_zero)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   714
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   715
lemma nonzero_add_divide_distrib: "c \<noteq> 0 ==> (a+b)/(c::'a::field) = a/c + b/c"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   716
by (simp add: divide_inverse left_distrib) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   717
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   718
lemma add_divide_distrib: "(a+b)/(c::'a::{field,division_by_zero}) = a/c + b/c"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   719
apply (case_tac "c=0", simp) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   720
apply (simp add: nonzero_add_divide_distrib) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   721
done
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   722
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   723
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
   724
      of an ordering.*}
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   725
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
   726
proof cases
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   727
  assume "a=0" thus ?thesis by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   728
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   729
  assume anz [simp]: "a\<noteq>0"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   730
  { assume "a * b = 0"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   731
    hence "inverse a * (a * b) = 0" by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   732
    hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   733
  thus ?thesis by force
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   734
qed
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   735
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   736
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
   737
lemma field_mult_cancel_right_lemma:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   738
      assumes cnz: "c \<noteq> (0::'a::field)"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   739
	  and eq:  "a*c = b*c"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   740
	 shows "a=b"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   741
proof -
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   742
  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
   743
    by (simp add: eq)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   744
  thus "a=b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   745
    by (simp add: mult_assoc cnz)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   746
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   747
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   748
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
   749
     "(a*c = b*c) = (c = (0::'a::field) | a=b)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   750
proof cases
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   751
  assume "c=0" thus ?thesis by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   752
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   753
  assume "c\<noteq>0" 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   754
  thus ?thesis by (force dest: field_mult_cancel_right_lemma)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   755
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   756
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   757
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
   758
     "(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
   759
  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
   760
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   761
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
   762
proof
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   763
  assume ianz: "inverse a = 0"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   764
  assume "a \<noteq> 0"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   765
  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
   766
  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
   767
  finally have "1 = (0::'a::field)" .
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   768
  thus False by (simp add: eq_commute)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   769
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   770
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   771
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   772
subsection{*Basic Properties of @{term inverse}*}
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   773
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   774
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
   775
apply (rule ccontr) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   776
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
   777
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   778
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   779
lemma inverse_nonzero_imp_nonzero:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   780
   "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
   781
apply (rule ccontr) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   782
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
   783
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   784
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   785
lemma inverse_nonzero_iff_nonzero [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   786
   "(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
   787
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
   788
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   789
lemma nonzero_inverse_minus_eq:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   790
      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
   791
proof -
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   792
  have "-a * inverse (- a) = -a * - inverse a"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   793
    by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   794
  thus ?thesis 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   795
    by (simp only: field_mult_cancel_left, simp)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   796
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   797
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   798
lemma inverse_minus_eq [simp]:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   799
   "inverse(-a) = -inverse(a::'a::{field,division_by_zero})";
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   800
proof cases
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   801
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   802
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   803
  assume "a\<noteq>0" 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   804
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   805
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   806
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   807
lemma nonzero_inverse_eq_imp_eq:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   808
      assumes inveq: "inverse a = inverse b"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   809
	  and anz:  "a \<noteq> 0"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   810
	  and bnz:  "b \<noteq> 0"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   811
	 shows "a = (b::'a::field)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   812
proof -
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   813
  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
   814
    by (simp add: inveq)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   815
  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
   816
    by simp
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   817
  thus "a = b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   818
    by (simp add: mult_assoc anz bnz)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   819
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   820
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   821
lemma inverse_eq_imp_eq:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   822
     "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
   823
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
   824
 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
   825
              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
   826
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
   827
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   828
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   829
lemma inverse_eq_iff_eq [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   830
     "(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
   831
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
   832
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   833
lemma nonzero_inverse_inverse_eq:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   834
      assumes [simp]: "a \<noteq> 0"  shows "inverse(inverse (a::'a::field)) = a"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   835
  proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   836
  have "(inverse (inverse a) * inverse a) * a = a" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   837
    by (simp add: nonzero_imp_inverse_nonzero)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   838
  thus ?thesis
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   839
    by (simp add: mult_assoc)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   840
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   841
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   842
lemma inverse_inverse_eq [simp]:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   843
     "inverse(inverse (a::'a::{field,division_by_zero})) = a"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   844
  proof cases
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   845
    assume "a=0" thus ?thesis by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   846
  next
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   847
    assume "a\<noteq>0" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   848
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   849
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   850
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   851
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   852
  proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   853
  have "inverse 1 * 1 = (1::'a::field)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   854
    by (rule left_inverse [OF zero_neq_one [symmetric]])
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   855
  thus ?thesis  by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   856
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   857
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   858
lemma nonzero_inverse_mult_distrib: 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   859
      assumes anz: "a \<noteq> 0"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   860
          and bnz: "b \<noteq> 0"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   861
      shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   862
  proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   863
  have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   864
    by (simp add: field_mult_eq_0_iff anz bnz)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   865
  hence "inverse(a*b) * a = inverse(b)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   866
    by (simp add: mult_assoc bnz)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   867
  hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   868
    by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   869
  thus ?thesis
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   870
    by (simp add: mult_assoc anz)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   871
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   872
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   873
text{*This version builds in division by zero while also re-orienting
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   874
      the right-hand side.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   875
lemma inverse_mult_distrib [simp]:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   876
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   877
  proof cases
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   878
    assume "a \<noteq> 0 & b \<noteq> 0" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   879
    thus ?thesis  by (simp add: nonzero_inverse_mult_distrib mult_commute)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   880
  next
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   881
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   882
    thus ?thesis  by force
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   883
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   884
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   885
text{*There is no slick version using division by zero.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   886
lemma inverse_add:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   887
     "[|a \<noteq> 0;  b \<noteq> 0|]
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   888
      ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   889
apply (simp add: left_distrib mult_assoc)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   890
apply (simp add: mult_commute [of "inverse a"]) 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   891
apply (simp add: mult_assoc [symmetric] add_commute)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   892
done
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   893
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   894
lemma inverse_divide [simp]:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   895
      "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   896
  by (simp add: divide_inverse_zero mult_commute)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   897
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   898
lemma nonzero_mult_divide_cancel_left:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   899
  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
   900
    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
   901
proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   902
  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
   903
    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
   904
                  nonzero_inverse_mult_distrib)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   905
  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
   906
    by (simp only: mult_ac)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   907
  also have "... =  a * inverse b"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   908
    by simp
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   909
    finally show ?thesis 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   910
    by (simp add: divide_inverse)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   911
qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   912
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   913
lemma mult_divide_cancel_left:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   914
     "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
   915
apply (case_tac "b = 0")
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   916
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
   917
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   918
14321
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   919
lemma nonzero_mult_divide_cancel_right:
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   920
     "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   921
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   922
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   923
lemma mult_divide_cancel_right:
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   924
     "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   925
apply (case_tac "b = 0")
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   926
apply (simp_all add: nonzero_mult_divide_cancel_right)
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   927
done
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   928
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   929
(*For ExtractCommonTerm*)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   930
lemma mult_divide_cancel_eq_if:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   931
     "(c*a) / (c*b) = 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   932
      (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
   933
  by (simp add: mult_divide_cancel_left)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   934
14284
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
   935
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
   936
  by (simp add: divide_inverse [OF not_sym])
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
   937
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   938
lemma times_divide_eq_right [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   939
     "a * (b/c) = (a*b) / (c::'a::{field,division_by_zero})"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   940
by (simp add: divide_inverse_zero mult_assoc)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   941
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   942
lemma times_divide_eq_left [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   943
     "(b/c) * a = (b*a) / (c::'a::{field,division_by_zero})"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   944
by (simp add: divide_inverse_zero mult_ac)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   945
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   946
lemma divide_divide_eq_right [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   947
     "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   948
by (simp add: divide_inverse_zero mult_ac)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   949
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   950
lemma divide_divide_eq_left [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   951
     "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   952
by (simp add: divide_inverse_zero mult_assoc)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   953
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   954
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   955
subsection {* Division and Unary Minus *}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   956
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   957
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
   958
by (simp add: divide_inverse minus_mult_left)
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   959
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   960
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
   961
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
   962
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   963
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
   964
by (simp add: divide_inverse nonzero_inverse_minus_eq)
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   965
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   966
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::{field,division_by_zero})"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   967
apply (case_tac "b=0", simp) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   968
apply (simp add: nonzero_minus_divide_left) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   969
done
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   970
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   971
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   972
apply (case_tac "b=0", simp) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   973
by (rule nonzero_minus_divide_right) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   974
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   975
text{*The effect is to extract signs from divisions*}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   976
declare minus_divide_left  [symmetric, simp]
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   977
declare minus_divide_right [symmetric, simp]
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   978
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   979
lemma minus_divide_divide [simp]:
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   980
     "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   981
apply (case_tac "b=0", simp) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   982
apply (simp add: nonzero_minus_divide_divide) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   983
done
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   984
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   985
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   986
subsection {* Ordered Fields *}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   987
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   988
lemma positive_imp_inverse_positive: 
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   989
      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
   990
  proof -
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   991
  have "0 < a * inverse a" 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   992
    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
   993
  thus "0 < inverse a" 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   994
    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
   995
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   996
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   997
lemma negative_imp_inverse_negative:
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   998
     "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
   999
  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
  1000
      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
  1001
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1002
lemma inverse_le_imp_le:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1003
      assumes invle: "inverse a \<le> inverse b"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1004
	  and apos:  "0 < a"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1005
	 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
  1006
  proof (rule classical)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1007
  assume "~ b \<le> a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1008
  hence "a < b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1009
    by (simp add: linorder_not_le)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1010
  hence bpos: "0 < b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1011
    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
  1012
  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
  1013
    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
  1014
  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
  1015
    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
  1016
  thus "b \<le> a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1017
    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
  1018
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1019
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1020
lemma inverse_positive_imp_positive:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1021
      assumes inv_gt_0: "0 < inverse a"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1022
          and [simp]:   "a \<noteq> 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1023
        shows "0 < (a::'a::ordered_field)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1024
  proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1025
  have "0 < inverse (inverse a)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1026
    by (rule positive_imp_inverse_positive)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1027
  thus "0 < a"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1028
    by (simp add: nonzero_inverse_inverse_eq)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1029
  qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1030
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1031
lemma inverse_positive_iff_positive [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1032
      "(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
  1033
apply (case_tac "a = 0", simp)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1034
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
  1035
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1036
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1037
lemma inverse_negative_imp_negative:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1038
      assumes inv_less_0: "inverse a < 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1039
          and [simp]:   "a \<noteq> 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1040
        shows "a < (0::'a::ordered_field)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1041
  proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1042
  have "inverse (inverse a) < 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1043
    by (rule negative_imp_inverse_negative)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1044
  thus "a < 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1045
    by (simp add: nonzero_inverse_inverse_eq)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1046
  qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1047
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1048
lemma inverse_negative_iff_negative [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1049
      "(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
  1050
apply (case_tac "a = 0", simp)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1051
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
  1052
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1053
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1054
lemma inverse_nonnegative_iff_nonnegative [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1055
      "(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
  1056
by (simp add: linorder_not_less [symmetric])
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1057
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1058
lemma inverse_nonpositive_iff_nonpositive [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1059
      "(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
  1060
by (simp add: linorder_not_less [symmetric])
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1061
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1062
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1063
subsection{*Anti-Monotonicity of @{term inverse}*}
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1064
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1065
lemma less_imp_inverse_less:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1066
      assumes less: "a < b"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1067
	  and apos:  "0 < a"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1068
	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
  1069
  proof (rule ccontr)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1070
  assume "~ inverse b < inverse a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1071
  hence "inverse a \<le> inverse b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1072
    by (simp add: linorder_not_less)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1073
  hence "~ (a < b)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1074
    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
  1075
  thus False
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1076
    by (rule notE [OF _ less])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1077
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1078
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1079
lemma inverse_less_imp_less:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1080
   "[|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
  1081
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
  1082
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
  1083
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1084
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1085
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
  1086
lemma inverse_less_iff_less [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1087
     "[|0 < a; 0 < b|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1088
      ==> (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
  1089
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
  1090
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1091
lemma le_imp_inverse_le:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1092
   "[|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
  1093
  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
  1094
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1095
lemma inverse_le_iff_le [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1096
     "[|0 < a; 0 < b|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1097
      ==> (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
  1098
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
  1099
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1100
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1101
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
  1102
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
  1103
lemma inverse_le_imp_le_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1104
   "[|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
  1105
  apply (rule classical) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1106
  apply (subgoal_tac "a < 0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1107
   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
  1108
  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
  1109
  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
  1110
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1111
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1112
lemma less_imp_inverse_less_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1113
   "[|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
  1114
  apply (subgoal_tac "a < 0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1115
   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
  1116
  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
  1117
  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
  1118
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1119
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1120
lemma inverse_less_imp_less_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1121
   "[|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
  1122
  apply (rule classical) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1123
  apply (subgoal_tac "a < 0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1124
   prefer 2
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1125
   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
  1126
  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
  1127
  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
  1128
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1129
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1130
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
  1131
     "[|a < 0; b < 0|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1132
      ==> (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
  1133
  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
  1134
  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
  1135
	      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
  1136
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1137
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1138
lemma le_imp_inverse_le_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1139
   "[|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
  1140
  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
  1141
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1142
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
  1143
     "[|a < 0; b < 0|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1144
      ==> (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
  1145
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
  1146
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1147
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1148
subsection{*Inverses and the Number One*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1149
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1150
lemma one_less_inverse_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1151
    "(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
  1152
  assume "0 < x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1153
    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
  1154
    show ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1155
next
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1156
  assume notless: "~ (0 < x)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1157
  have "~ (1 < inverse x)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1158
  proof
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1159
    assume "1 < inverse x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1160
    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
  1161
    also have "... < 1" by (rule zero_less_one) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1162
    finally show False by auto
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1163
  qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1164
  with notless show ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1165
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1166
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1167
lemma inverse_eq_1_iff [simp]:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1168
    "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1169
by (insert inverse_eq_iff_eq [of x 1], simp) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1170
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1171
lemma one_le_inverse_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1172
   "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1173
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1174
                    eq_commute [of 1]) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1175
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1176
lemma inverse_less_1_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1177
   "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1178
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1179
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1180
lemma inverse_le_1_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1181
   "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1182
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1183
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1184
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1185
subsection{*Division and Signs*}
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1186
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1187
lemma zero_less_divide_iff:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1188
     "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1189
by (simp add: divide_inverse_zero zero_less_mult_iff)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1190
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1191
lemma divide_less_0_iff:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1192
     "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1193
      (0 < a & b < 0 | a < 0 & 0 < b)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1194
by (simp add: divide_inverse_zero mult_less_0_iff)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1195
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1196
lemma zero_le_divide_iff:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1197
     "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1198
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1199
by (simp add: divide_inverse_zero zero_le_mult_iff)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1200
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1201
lemma divide_le_0_iff:
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1202
     "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1203
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1204
by (simp add: divide_inverse_zero mult_le_0_iff)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1205
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1206
lemma divide_eq_0_iff [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1207
     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1208
by (simp add: divide_inverse_zero field_mult_eq_0_iff)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1209
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1210
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1211
subsection{*Simplification of Inequalities Involving Literal Divisors*}
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1212
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1213
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1214
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1215
  assume less: "0<c"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1216
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1217
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1218
  also have "... = (a*c \<le> b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1219
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1220
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1221
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1222
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1223
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1224
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1225
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1226
  assume less: "c<0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1227
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1228
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1229
  also have "... = (b \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1230
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1231
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1232
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1233
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1234
lemma le_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1235
  "(a \<le> b/c) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1236
   (if 0 < c then a*c \<le> b
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1237
             else if c < 0 then b \<le> a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1238
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1239
apply (case_tac "c=0", simp) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1240
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1241
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1242
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1243
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1244
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1245
  assume less: "0<c"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1246
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1247
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1248
  also have "... = (b \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1249
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1250
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1251
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1252
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1253
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1254
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1255
  assume less: "c<0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1256
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1257
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1258
  also have "... = (a*c \<le> b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1259
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1260
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1261
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1262
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1263
lemma divide_le_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1264
  "(b/c \<le> a) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1265
   (if 0 < c then b \<le> a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1266
             else if c < 0 then a*c \<le> b
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1267
             else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1268
apply (case_tac "c=0", simp) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1269
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1270
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1271
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1272
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1273
lemma pos_less_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1274
     "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1275
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1276
  assume less: "0<c"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1277
  hence "(a < b/c) = (a*c < (b/c)*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1278
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1279
  also have "... = (a*c < b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1280
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1281
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1282
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1283
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1284
lemma neg_less_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1285
 "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1286
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1287
  assume less: "c<0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1288
  hence "(a < b/c) = ((b/c)*c < a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1289
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1290
  also have "... = (b < a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1291
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1292
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1293
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1294
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1295
lemma less_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1296
  "(a < b/c) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1297
   (if 0 < c then a*c < b
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1298
             else if c < 0 then b < a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1299
             else  a < (0::'a::{ordered_field,division_by_zero}))"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1300
apply (case_tac "c=0", simp) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1301
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset