src/HOL/Ring_and_Field.thy
author paulson
Tue, 06 Jan 2004 10:40:15 +0100
changeset 14341 a09441bd4f1e
parent 14334 6137d24eef79
child 14348 744c868ee0b7
permissions -rw-r--r--
Ring_and_Field now requires axiom add_left_imp_eq for semirings. This allows more theorems to be proved for semirings, but requires a redundant axiom to be proved for rings, etc.
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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 and Markus Wenzel}
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*}
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theory Ring_and_Field = Inductive:
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text{*Lemmas and extension to semirings by L. C. Paulson*}
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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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  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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diff changeset
   135
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
   136
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   137
lemma neg_0_equal_iff_equal [simp]: "(0 = -a) = (0 = (a::'a::ring))"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   138
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
   139
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   140
text{*The next two equations can make the simplifier loop!*}
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   141
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   142
lemma equation_minus_iff: "(a = - b) = (b = - (a::'a::ring))"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   143
  proof -
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   144
  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
   145
  thus ?thesis by (simp add: eq_commute)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   146
  qed
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   147
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   148
lemma minus_equation_iff: "(- a = b) = (- (b::'a::ring) = a)"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   149
  proof -
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   150
  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
   151
  thus ?thesis by (simp add: eq_commute)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   152
  qed
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   153
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   154
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   155
subsection {* Derived rules for multiplication *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   156
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   157
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
   158
proof -
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   159
  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
   160
  also have "... = a" by simp
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   161
  finally show ?thesis .
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   162
qed
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   163
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   164
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
   165
  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
   166
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   167
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
   168
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   169
lemma mult_left_zero [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
   170
proof -
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   171
  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
   172
    by (simp add: left_distrib [symmetric])
14266
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
   173
  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
   174
qed
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   175
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   176
lemma mult_right_zero [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
   177
  by (simp add: mult_commute)
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
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   180
subsection {* Distribution rules *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   181
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   182
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
   183
proof -
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   184
  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
   185
  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
   186
  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
   187
  finally show ?thesis .
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   188
qed
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   189
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   190
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
   191
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   192
text{*For the @{text combine_numerals} simproc*}
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   193
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
   194
by (simp add: left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   195
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   196
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
   197
apply (rule equals_zero_I)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   198
apply (simp add: add_ac) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   199
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   200
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   201
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
   202
apply (rule equals_zero_I)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   203
apply (simp add: left_distrib [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   204
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   205
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   206
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
   207
apply (rule equals_zero_I)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   208
apply (simp add: right_distrib [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   209
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   210
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   211
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
   212
  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
   213
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   214
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
   215
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
   216
              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
   217
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   218
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
   219
by (simp add: mult_commute [of _ c] right_diff_distrib) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   220
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   221
lemma minus_diff_eq [simp]: "- (a - b) = b - (a::'a::ring)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   222
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
   223
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   224
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   225
subsection {* Ordering Rules for Addition *}
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   226
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   227
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
   228
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
   229
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   230
text {* non-strict, in both arguments *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   231
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
   232
  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
   233
  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
   234
  done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   235
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   236
lemma add_strict_left_mono:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   237
     "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
   238
 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
   239
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   240
lemma add_strict_right_mono:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   241
     "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
   242
 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
   243
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   244
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
   245
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
   246
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
   247
apply (erule add_strict_left_mono)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   248
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   249
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   250
lemma add_less_le_mono: "[| a<b; c\<le>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
   251
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
   252
apply (erule add_left_mono) 
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   253
done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   254
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   255
lemma add_le_less_mono:
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   256
     "[| 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
   257
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
   258
apply (erule add_strict_left_mono) 
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   259
done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   260
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   261
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
   262
      assumes less: "c + a < c + b"  shows "a < (b::'a::ordered_semiring)"
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   263
  proof (rule ccontr)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   264
    assume "~ a < b"
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   265
    hence "b \<le> a" by (simp add: linorder_not_less)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   266
    hence "c+b \<le> c+a" by (rule add_left_mono)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   267
    with this and less show False 
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   268
      by (simp add: linorder_not_less [symmetric])
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   269
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   271
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
   272
      "a + c < b + c ==> a < (b::'a::ordered_semiring)"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   273
apply (rule add_less_imp_less_left [of c])
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   274
apply (simp add: add_commute)  
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   275
done
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   276
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   277
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
   278
    "(c+a < c+b) = (a < (b::'a::ordered_semiring))"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   279
by (blast intro: add_less_imp_less_left add_strict_left_mono) 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   280
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   281
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
   282
    "(a+c < b+c) = (a < (b::'a::ordered_semiring))"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   283
by (blast intro: add_less_imp_less_right add_strict_right_mono)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   284
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   285
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
   286
    "(c+a \<le> c+b) = (a \<le> (b::'a::ordered_semiring))"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   287
by (simp add: linorder_not_less [symmetric]) 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   288
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   289
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
   290
    "(a+c \<le> b+c) = (a \<le> (b::'a::ordered_semiring))"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   291
by (simp add: linorder_not_less [symmetric]) 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   292
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   293
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
   294
      "c + a \<le> c + b ==> a \<le> (b::'a::ordered_semiring)"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   295
by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   296
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   297
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
   298
      "a + c \<le> b + c ==> a \<le> (b::'a::ordered_semiring)"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   299
by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   300
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   301
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   302
subsection {* Ordering Rules for Unary Minus *}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   303
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   304
lemma le_imp_neg_le:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   305
      assumes "a \<le> (b::'a::ordered_ring)" shows "-b \<le> -a"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   306
  proof -
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   307
  have "-a+a \<le> -a+b"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   308
    by (rule add_left_mono) 
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   309
  hence "0 \<le> -a+b"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   310
    by simp
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   311
  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
   312
    by (rule add_right_mono) 
14266
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
   313
  thus ?thesis
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   314
    by (simp add: add_assoc)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   315
  qed
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   316
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   317
lemma neg_le_iff_le [simp]: "(-b \<le> -a) = (a \<le> (b::'a::ordered_ring))"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   318
  proof 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   319
    assume "- b \<le> - a"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   320
    hence "- (- a) \<le> - (- b)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   321
      by (rule le_imp_neg_le)
14266
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
   322
    thus "a\<le>b" by simp
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   323
  next
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   324
    assume "a\<le>b"
14266
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
   325
    thus "-b \<le> -a" by (rule le_imp_neg_le)
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   326
  qed
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   327
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   328
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
   329
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
   330
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   331
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
   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_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
   335
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
   336
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   337
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
   338
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
   339
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   340
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
   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
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   343
text{*The next several equations can make the simplifier loop!*}
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   344
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   345
lemma less_minus_iff: "(a < - b) = (b < - (a::'a::ordered_ring))"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   346
  proof -
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   347
  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
   348
  thus ?thesis by simp
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   349
  qed
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   350
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   351
lemma minus_less_iff: "(- a < b) = (- b < (a::'a::ordered_ring))"
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   352
  proof -
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   353
  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
   354
  thus ?thesis by simp
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   355
  qed
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   356
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   357
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
   358
apply (simp add: linorder_not_less [symmetric])
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   359
apply (rule minus_less_iff) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   360
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   361
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   362
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
   363
apply (simp add: linorder_not_less [symmetric])
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   364
apply (rule less_minus_iff) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   365
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   366
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   367
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   368
subsection{*Subtraction Laws*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   369
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   370
lemma add_diff_eq: "a + (b - c) = (a + b) - (c::'a::ring)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   371
by (simp add: diff_minus add_ac)
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 diff_add_eq: "(a - b) + c = (a + c) - (b::'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_eq_eq: "(a-b = c) = (a = c + (b::'a::ring))"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   377
by (auto simp add: diff_minus add_assoc)
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 eq_diff_eq: "(a = c-b) = (a + (b::'a::ring) = c)"
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 diff_diff_eq: "(a - b) - c = a - (b + (c::'a::ring))"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   383
by (simp add: diff_minus add_ac)
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_eq2: "a - (b - c) = (a + c) - (b::'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
text{*Further subtraction laws for ordered rings*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   389
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   390
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
   391
proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   392
  have  "(a < b) = (a + (- b) < b + (-b))"  
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   393
    by (simp only: add_less_cancel_right)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   394
  also have "... =  (a - b < 0)" by (simp add: diff_minus)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   395
  finally show ?thesis .
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   396
qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   397
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   398
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
   399
apply (subst less_iff_diff_less_0)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   400
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   401
apply (simp add: diff_minus add_ac)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   402
done
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   403
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   404
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
   405
apply (subst less_iff_diff_less_0)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   406
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
   407
apply (simp add: diff_minus add_ac)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   408
done
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   409
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   410
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
   411
by (simp add: linorder_not_less [symmetric] less_diff_eq)
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 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
   414
by (simp add: linorder_not_less [symmetric] diff_less_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
text{*This list of rewrites simplifies (in)equalities by bringing subtractions
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   417
  to the top and then moving negative terms to the other side.
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   418
  Use with @{text add_ac}*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   419
lemmas compare_rls =
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   420
       diff_minus [symmetric]
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   421
       add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   422
       diff_less_eq less_diff_eq diff_le_eq le_diff_eq
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   423
       diff_eq_eq eq_diff_eq
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   424
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   425
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   426
subsection{*Lemmas for the @{text cancel_numerals} simproc*}
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   427
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   428
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
   429
by (simp add: compare_rls)
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 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
   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 eq_add_iff1:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   435
     "(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
   436
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   437
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   438
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   439
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   440
lemma eq_add_iff2:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   441
     "(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
   442
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   443
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   444
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   445
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   446
lemma less_add_iff1:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   447
     "(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
   448
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   449
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   450
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   451
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   452
lemma less_add_iff2:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   453
     "(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
   454
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   455
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   456
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   457
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   458
lemma le_add_iff1:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   459
     "(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
   460
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   461
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   462
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   463
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   464
lemma le_add_iff2:
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   465
     "(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
   466
apply (simp add: diff_minus left_distrib add_ac)
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   467
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   468
done
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   469
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   470
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   471
subsection {* Ordering Rules for Multiplication *}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   472
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   473
lemma mult_strict_right_mono:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   474
     "[|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
   475
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
   476
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   477
lemma mult_left_mono:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   478
     "[|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
   479
  apply (case_tac "c=0", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   480
  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
   481
  done
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   482
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   483
lemma mult_right_mono:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   484
     "[|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
   485
  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
   486
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   487
lemma mult_strict_left_mono_neg:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   488
     "[|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
   489
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
   490
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
   491
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   492
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   493
lemma mult_strict_right_mono_neg:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   494
     "[|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
   495
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
   496
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
   497
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   498
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   499
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   500
subsection{* Products of Signs *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   501
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   502
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
   503
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
   504
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   505
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
   506
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
   507
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   508
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
   509
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
   510
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   511
lemma zero_less_mult_pos:
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   512
     "[| 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
   513
apply (case_tac "b\<le>0") 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   514
 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
   515
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
   516
 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
   517
done
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
lemma zero_less_mult_iff:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   520
     "((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
   521
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
   522
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
   523
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
   524
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
   525
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   526
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   527
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
   528
      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
   529
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
   530
apply (case_tac "a < 0")
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   531
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
   532
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
   533
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   534
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   535
lemma zero_le_mult_iff:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   536
     "((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
   537
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
   538
                   zero_less_mult_iff)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   539
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   540
lemma mult_less_0_iff:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   541
     "(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
   542
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
   543
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
   544
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   545
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   546
lemma mult_le_0_iff:
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   547
     "(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
   548
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
   549
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
   550
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   551
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   552
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
   553
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
   554
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   555
lemma zero_less_one: "(0::'a::ordered_ring) < 1"
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   556
apply (insert zero_le_square [of 1]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   557
apply (simp add: order_less_le) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   558
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   559
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   560
lemma zero_le_one: "(0::'a::ordered_ring) \<le> 1"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   561
  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
   562
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   563
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   564
subsection{*More Monotonicity*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   565
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   566
lemma mult_left_mono_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   567
     "[|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
   568
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
   569
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
   570
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   571
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   572
lemma mult_right_mono_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   573
     "[|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
   574
  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
   575
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   576
text{*Strict monotonicity in both arguments*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   577
lemma mult_strict_mono:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   578
     "[|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
   579
apply (case_tac "c=0")
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   580
 apply (simp add: mult_pos) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   581
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
   582
 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
   583
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
   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
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
   587
lemma mult_strict_mono':
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   588
     "[| 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
   589
apply (rule mult_strict_mono)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   590
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
   591
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   592
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   593
lemma mult_mono:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   594
     "[|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
   595
      ==> 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
   596
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
   597
apply (erule mult_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
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   601
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
   602
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   603
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
   604
   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
   605
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   606
lemma mult_less_cancel_right:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   607
    "(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
   608
apply (case_tac "c = 0")
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   609
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
   610
                      mult_strict_right_mono_neg)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   611
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
   612
                      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
   613
                      linorder_not_le [symmetric, of a])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   614
apply (erule_tac [!] notE)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   615
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
   616
                      mult_right_mono_neg)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   617
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   618
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   619
lemma mult_less_cancel_left:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   620
    "(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
   621
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
   622
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   623
lemma mult_le_cancel_right:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   624
     "(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
   625
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
   626
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   627
lemma mult_le_cancel_left:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   628
     "(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
   629
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
   630
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   631
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
   632
      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
   633
      shows "a < (b::'a::ordered_semiring)"
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   634
  proof (rule ccontr)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   635
    assume "~ a < b"
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   636
    hence "b \<le> a" by (simp add: linorder_not_less)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   637
    hence "c*b \<le> c*a" by (rule mult_left_mono)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   638
    with this and less show False 
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   639
      by (simp add: linorder_not_less [symmetric])
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   640
  qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   641
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   642
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
   643
    "[|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
   644
  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
   645
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   646
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
   647
lemma mult_cancel_right [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   648
     "(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
   649
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
   650
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
   651
             simp add: linorder_neq_iff)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   652
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   653
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   654
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
   655
      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
   656
lemma mult_cancel_left [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   657
     "(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
   658
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
   659
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   660
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   661
subsection {* Fields *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   662
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   663
lemma right_inverse [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   664
      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
   665
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   666
  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
   667
  also have "... = 1" using not0 by simp
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   668
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   669
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   670
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   671
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
   672
proof
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   673
  assume neq: "b \<noteq> 0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   674
  {
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   675
    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
   676
    also assume "a / b = 1"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   677
    finally show "a = b" by simp
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   678
  next
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   679
    assume "a = b"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   680
    with neq show "a / b = 1" by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   681
  }
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   682
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   683
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   684
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
   685
by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   686
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   687
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
   688
  by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   689
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   690
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
   691
apply (case_tac "b = 0")
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   692
apply (simp_all add: divide_inverse)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   693
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   694
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   695
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
   696
by (simp add: divide_inverse_zero)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   697
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   698
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
   699
by (simp add: divide_inverse_zero)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   700
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   701
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
   702
by (simp add: divide_inverse left_distrib) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   703
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   704
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
   705
apply (case_tac "c=0", simp) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   706
apply (simp add: nonzero_add_divide_distrib) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   707
done
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   708
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   709
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
   710
      of an ordering.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   711
lemma field_mult_eq_0_iff: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   712
  proof cases
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   713
    assume "a=0" thus ?thesis by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   714
  next
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   715
    assume anz [simp]: "a\<noteq>0"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   716
    thus ?thesis
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   717
    proof auto
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   718
      assume "a * b = 0"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   719
      hence "inverse a * (a * b) = 0" by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   720
      thus "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   721
    qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   722
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   723
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   724
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
   725
lemma field_mult_cancel_right_lemma:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   726
      assumes cnz: "c \<noteq> (0::'a::field)"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   727
	  and eq:  "a*c = b*c"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   728
	 shows "a=b"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   729
  proof -
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   730
  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
   731
    by (simp add: eq)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   732
  thus "a=b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   733
    by (simp add: mult_assoc cnz)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   734
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   735
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   736
lemma field_mult_cancel_right:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   737
     "(a*c = b*c) = (c = (0::'a::field) | a=b)"
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   738
  proof cases
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   739
    assume "c=0" thus ?thesis by simp
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   740
  next
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   741
    assume "c\<noteq>0" 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   742
    thus ?thesis by (force dest: field_mult_cancel_right_lemma)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   743
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   744
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   745
lemma field_mult_cancel_left:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   746
     "(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
   747
  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
   748
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   749
lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   750
  proof
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   751
  assume ianz: "inverse a = 0"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   752
  assume "a \<noteq> 0"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   753
  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
   754
  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
   755
  finally have "1 = (0::'a::field)" .
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   756
  thus False by (simp add: eq_commute)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   757
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   758
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   759
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   760
subsection{*Basic Properties of @{term inverse}*}
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   761
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   762
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
   763
apply (rule ccontr) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   764
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
   765
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   766
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   767
lemma inverse_nonzero_imp_nonzero:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   768
   "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
   769
apply (rule ccontr) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   770
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
   771
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   772
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   773
lemma inverse_nonzero_iff_nonzero [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   774
   "(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
   775
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
   776
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   777
lemma nonzero_inverse_minus_eq:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   778
      assumes [simp]: "a\<noteq>0"  shows "inverse(-a) = -inverse(a::'a::field)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   779
  proof -
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   780
    have "-a * inverse (- a) = -a * - inverse a"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   781
      by simp
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   782
    thus ?thesis 
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   783
      by (simp only: field_mult_cancel_left, simp)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   784
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   785
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   786
lemma inverse_minus_eq [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   787
     "inverse(-a) = -inverse(a::'a::{field,division_by_zero})";
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   788
  proof cases
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   789
    assume "a=0" thus ?thesis by (simp add: inverse_zero)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   790
  next
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   791
    assume "a\<noteq>0" 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   792
    thus ?thesis by (simp add: nonzero_inverse_minus_eq)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   793
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   794
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   795
lemma nonzero_inverse_eq_imp_eq:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   796
      assumes inveq: "inverse a = inverse b"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   797
	  and anz:  "a \<noteq> 0"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   798
	  and bnz:  "b \<noteq> 0"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   799
	 shows "a = (b::'a::field)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   800
  proof -
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   801
  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
   802
    by (simp add: inveq)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   803
  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
   804
    by simp
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   805
  thus "a = b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   806
    by (simp add: mult_assoc anz bnz)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   807
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   808
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   809
lemma inverse_eq_imp_eq:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   810
     "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
   811
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
   812
 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
   813
              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
   814
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
   815
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   816
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   817
lemma inverse_eq_iff_eq [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   818
     "(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
   819
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
   820
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   821
lemma nonzero_inverse_inverse_eq:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   822
      assumes [simp]: "a \<noteq> 0"  shows "inverse(inverse (a::'a::field)) = a"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   823
  proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   824
  have "(inverse (inverse a) * inverse a) * a = a" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   825
    by (simp add: nonzero_imp_inverse_nonzero)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   826
  thus ?thesis
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   827
    by (simp add: mult_assoc)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   828
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   829
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   830
lemma inverse_inverse_eq [simp]:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   831
     "inverse(inverse (a::'a::{field,division_by_zero})) = a"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   832
  proof cases
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   833
    assume "a=0" thus ?thesis by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   834
  next
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   835
    assume "a\<noteq>0" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   836
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   837
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   838
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   839
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   840
  proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   841
  have "inverse 1 * 1 = (1::'a::field)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   842
    by (rule left_inverse [OF zero_neq_one [symmetric]])
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   843
  thus ?thesis  by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   844
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   845
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   846
lemma nonzero_inverse_mult_distrib: 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   847
      assumes anz: "a \<noteq> 0"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   848
          and bnz: "b \<noteq> 0"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   849
      shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   850
  proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   851
  have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   852
    by (simp add: field_mult_eq_0_iff anz bnz)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   853
  hence "inverse(a*b) * a = inverse(b)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   854
    by (simp add: mult_assoc bnz)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   855
  hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   856
    by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   857
  thus ?thesis
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   858
    by (simp add: mult_assoc anz)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   859
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   860
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   861
text{*This version builds in division by zero while also re-orienting
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   862
      the right-hand side.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   863
lemma inverse_mult_distrib [simp]:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   864
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   865
  proof cases
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   866
    assume "a \<noteq> 0 & b \<noteq> 0" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   867
    thus ?thesis  by (simp add: nonzero_inverse_mult_distrib mult_commute)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   868
  next
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   869
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   870
    thus ?thesis  by force
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{*There is no slick version using division by zero.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   874
lemma inverse_add:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   875
     "[|a \<noteq> 0;  b \<noteq> 0|]
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   876
      ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   877
apply (simp add: left_distrib mult_assoc)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   878
apply (simp add: mult_commute [of "inverse a"]) 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   879
apply (simp add: mult_assoc [symmetric] add_commute)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   880
done
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   881
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   882
lemma nonzero_mult_divide_cancel_left:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   883
  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
   884
    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
   885
proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   886
  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
   887
    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
   888
                  nonzero_inverse_mult_distrib)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   889
  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
   890
    by (simp only: mult_ac)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   891
  also have "... =  a * inverse b"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   892
    by simp
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   893
    finally show ?thesis 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   894
    by (simp add: divide_inverse)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   895
qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   896
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   897
lemma mult_divide_cancel_left:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   898
     "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
   899
apply (case_tac "b = 0")
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   900
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
   901
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   902
14321
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   903
lemma nonzero_mult_divide_cancel_right:
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   904
     "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   905
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   906
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   907
lemma mult_divide_cancel_right:
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   908
     "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   909
apply (case_tac "b = 0")
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   910
apply (simp_all add: nonzero_mult_divide_cancel_right)
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   911
done
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   912
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   913
(*For ExtractCommonTerm*)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   914
lemma mult_divide_cancel_eq_if:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   915
     "(c*a) / (c*b) = 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   916
      (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
   917
  by (simp add: mult_divide_cancel_left)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   918
14284
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
   919
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
   920
  by (simp add: divide_inverse [OF not_sym])
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
   921
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   922
lemma times_divide_eq_right [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   923
     "a * (b/c) = (a*b) / (c::'a::{field,division_by_zero})"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   924
by (simp add: divide_inverse_zero mult_assoc)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   925
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   926
lemma times_divide_eq_left [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   927
     "(b/c) * a = (b*a) / (c::'a::{field,division_by_zero})"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   928
by (simp add: divide_inverse_zero mult_ac)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   929
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   930
lemma divide_divide_eq_right [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   931
     "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   932
by (simp add: divide_inverse_zero mult_ac)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   933
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   934
lemma divide_divide_eq_left [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   935
     "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   936
by (simp add: divide_inverse_zero mult_assoc)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   937
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   938
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   939
subsection {* Division and Unary Minus *}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   940
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   941
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
   942
by (simp add: divide_inverse minus_mult_left)
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   943
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   944
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
   945
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
   946
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   947
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
   948
by (simp add: divide_inverse nonzero_inverse_minus_eq)
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   949
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   950
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
   951
apply (case_tac "b=0", simp) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   952
apply (simp add: nonzero_minus_divide_left) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   953
done
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   954
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   955
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
   956
apply (case_tac "b=0", simp) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   957
by (rule nonzero_minus_divide_right) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   958
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   959
text{*The effect is to extract signs from divisions*}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   960
declare minus_divide_left  [symmetric, simp]
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   961
declare minus_divide_right [symmetric, simp]
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 minus_divide_divide [simp]:
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   964
     "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   965
apply (case_tac "b=0", simp) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   966
apply (simp add: nonzero_minus_divide_divide) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   967
done
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   968
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   969
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   970
subsection {* Ordered Fields *}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   971
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   972
lemma positive_imp_inverse_positive: 
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   973
      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
   974
  proof -
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   975
  have "0 < a * inverse a" 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   976
    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
   977
  thus "0 < inverse a" 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   978
    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
   979
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   980
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   981
lemma negative_imp_inverse_negative:
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   982
     "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
   983
  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
   984
      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
   985
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   986
lemma inverse_le_imp_le:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   987
      assumes invle: "inverse a \<le> inverse b"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   988
	  and apos:  "0 < a"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   989
	 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
   990
  proof (rule classical)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   991
  assume "~ b \<le> a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   992
  hence "a < b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   993
    by (simp add: linorder_not_le)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   994
  hence bpos: "0 < b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   995
    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
   996
  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
   997
    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
   998
  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
   999
    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
  1000
  thus "b \<le> a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1001
    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
  1002
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1003
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1004
lemma inverse_positive_imp_positive:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1005
      assumes inv_gt_0: "0 < inverse a"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1006
          and [simp]:   "a \<noteq> 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1007
        shows "0 < (a::'a::ordered_field)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1008
  proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1009
  have "0 < inverse (inverse a)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1010
    by (rule positive_imp_inverse_positive)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1011
  thus "0 < a"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1012
    by (simp add: nonzero_inverse_inverse_eq)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1013
  qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1014
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1015
lemma inverse_positive_iff_positive [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1016
      "(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
  1017
apply (case_tac "a = 0", simp)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1018
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
  1019
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1020
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1021
lemma inverse_negative_imp_negative:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1022
      assumes inv_less_0: "inverse a < 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1023
          and [simp]:   "a \<noteq> 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1024
        shows "a < (0::'a::ordered_field)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1025
  proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1026
  have "inverse (inverse a) < 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1027
    by (rule negative_imp_inverse_negative)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1028
  thus "a < 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1029
    by (simp add: nonzero_inverse_inverse_eq)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1030
  qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1031
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1032
lemma inverse_negative_iff_negative [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1033
      "(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
  1034
apply (case_tac "a = 0", simp)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1035
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
  1036
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1037
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1038
lemma inverse_nonnegative_iff_nonnegative [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1039
      "(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
  1040
by (simp add: linorder_not_less [symmetric])
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1041
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1042
lemma inverse_nonpositive_iff_nonpositive [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1043
      "(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
  1044
by (simp add: linorder_not_less [symmetric])
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1045
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1046
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1047
subsection{*Anti-Monotonicity of @{term inverse}*}
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1048
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1049
lemma less_imp_inverse_less:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1050
      assumes less: "a < b"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1051
	  and apos:  "0 < a"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1052
	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
  1053
  proof (rule ccontr)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1054
  assume "~ inverse b < inverse a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1055
  hence "inverse a \<le> inverse b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1056
    by (simp add: linorder_not_less)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1057
  hence "~ (a < b)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1058
    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
  1059
  thus False
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1060
    by (rule notE [OF _ less])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1061
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1062
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1063
lemma inverse_less_imp_less:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1064
   "[|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
  1065
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
  1066
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
  1067
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1068
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1069
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
  1070
lemma inverse_less_iff_less [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1071
     "[|0 < a; 0 < b|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1072
      ==> (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
  1073
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
  1074
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1075
lemma le_imp_inverse_le:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1076
   "[|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
  1077
  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
  1078
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1079
lemma inverse_le_iff_le [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1080
     "[|0 < a; 0 < b|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1081
      ==> (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
  1082
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
  1083
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{*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
  1086
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
  1087
lemma inverse_le_imp_le_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1088
   "[|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
  1089
  apply (rule classical) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1090
  apply (subgoal_tac "a < 0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1091
   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
  1092
  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
  1093
  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
  1094
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1095
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1096
lemma less_imp_inverse_less_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1097
   "[|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
  1098
  apply (subgoal_tac "a < 0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1099
   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
  1100
  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
  1101
  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
  1102
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1103
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1104
lemma inverse_less_imp_less_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1105
   "[|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
  1106
  apply (rule classical) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1107
  apply (subgoal_tac "a < 0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1108
   prefer 2
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1109
   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
  1110
  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
  1111
  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
  1112
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1113
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1114
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
  1115
     "[|a < 0; b < 0|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1116
      ==> (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
  1117
  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
  1118
  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
  1119
	      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
  1120
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1121
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1122
lemma le_imp_inverse_le_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1123
   "[|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
  1124
  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
  1125
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1126
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
  1127
     "[|a < 0; b < 0|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1128
      ==> (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
  1129
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
  1130
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1131
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1132
subsection{*Division and Signs*}
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1133
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1134
lemma zero_less_divide_iff:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1135
     "((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
  1136
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
  1137
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1138
lemma divide_less_0_iff:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1139
     "(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
  1140
      (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
  1141
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
  1142
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1143
lemma zero_le_divide_iff:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1144
     "((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
  1145
      (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
  1146
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
  1147
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1148
lemma divide_le_0_iff:
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1149
     "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1150
      (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
  1151
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
  1152
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1153
lemma divide_eq_0_iff [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1154
     "(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
  1155
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
  1156
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1157
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1158
subsection{*Simplification of Inequalities Involving Literal Divisors*}
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1159
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1160
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
  1161
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1162
  assume less: "0<c"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1163
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1164
    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
  1165
  also have "... = (a*c \<le> b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1166
    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
  1167
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1168
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1169
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1170
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1171
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
  1172
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1173
  assume less: "c<0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1174
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1175
    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
  1176
  also have "... = (b \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1177
    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
  1178
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1179
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1180
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1181
lemma le_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1182
  "(a \<le> b/c) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1183
   (if 0 < c then a*c \<le> b
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1184
             else if c < 0 then b \<le> a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1185
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1186
apply (case_tac "c=0", simp) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1187
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
  1188
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1189
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1190
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
  1191
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1192
  assume less: "0<c"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1193
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1194
    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
  1195
  also have "... = (b \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1196
    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
  1197
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1198
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1199
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1200
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
  1201
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1202
  assume less: "c<0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1203
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1204
    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
  1205
  also have "... = (a*c \<le> b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1206
    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
  1207
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1208
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1209
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1210
lemma divide_le_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1211
  "(b/c \<le> a) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1212
   (if 0 < c then b \<le> a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1213
             else if c < 0 then a*c \<le> b
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1214
             else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1215
apply (case_tac "c=0", simp) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1216
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
  1217
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1218
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1219
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1220
lemma pos_less_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1221
     "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1222
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1223
  assume less: "0<c"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1224
  hence "(a < b/c) = (a*c < (b/c)*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1225
    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
  1226
  also have "... = (a*c < b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1227
    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
  1228
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1229
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1230
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1231
lemma neg_less_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1232
 "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1233
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1234
  assume less: "c<0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1235
  hence "(a < b/c) = ((b/c)*c < a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1236
    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
  1237
  also have "... = (b < a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1238
    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
  1239
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1240
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1241
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1242
lemma less_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1243
  "(a < b/c) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1244
   (if 0 < c then a*c < b
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1245
             else if c < 0 then b < a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1246
             else  a < (0::'a::{ordered_field,division_by_zero}))"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1247
apply (case_tac "c=0", simp) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1248
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
  1249
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1250
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1251
lemma pos_divide_less_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1252
     "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1253
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1254
  assume less: "0<c"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1255
  hence "(b/c < a) = ((b/c)*c < a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1256
    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
  1257
  also have "... = (b < a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1258
    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
  1259
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1260
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1261
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1262
lemma neg_divide_less_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1263
 "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1264
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1265
  assume less: "c<0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1266
  hence "(b/c < a) = (a*c < (b/c)*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1267
    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
  1268
  also have "... = (a*c < b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1269
    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
  1270
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1271
qed
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 divide_less_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1274
  "(b/c < a) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1275
   (if 0 < c then b < a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1276
             else if c < 0 then a*c < b
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1277
             else 0 < (a::'a::{ordered_field,division_by_zero}))"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1278
apply (case_tac "c=0", simp) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1279
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1280
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1281
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1282
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1283
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1284
  assume [simp]: "c\<noteq>0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1285
  have "(a = b/c) = (a*c = (b/c)*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1286
    by (simp add: field_mult_cancel_right)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1287
  also have "... = (a*c = b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1288
    by (simp add: divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1289
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1290
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1291
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1292
lemma eq_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1293
  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1294
by (simp add: nonzero_eq_divide_eq) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1295
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1296
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1297
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1298
  assume [simp]: "c\<noteq>0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1299
  have "(b/c = a) = ((b/c)*c = a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1300
    by (simp add: field_mult_cancel_right)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1301
  also have "... = (b = a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1302
    by (simp add: divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1303
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1304
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1305
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1306
lemma divide_eq_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1307
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1308
by (force simp add: nonzero_divide_eq_eq) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1309
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML