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permissions  rwrr 
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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 
14269  4 
Lawrence C Paulson, University of Cambridge 
14738  5 
Revised and splitted into Ring_and_Field.thy and Group.thy 
6 
by Steven Obua, TU Muenchen, in May 2004 

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License: GPL (GNU GENERAL PUBLIC LICENSE) 
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*) 
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14738  10 
header {* (Ordered) Rings and Fields *} 
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14738  12 
theory Ring_and_Field = OrderedGroup: 
14504  13 

14738  14 
text {* 
15 
The theory of partially ordered rings is taken from the books: 

16 
\begin{itemize} 

17 
\item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 

18 
\item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963 

19 
\end{itemize} 

20 
Most of the used notions can also be looked up in 

21 
\begin{itemize} 

22 
\item \emph{www.mathworld.com} by Eric Weisstein et. al. 

23 
\item \emph{Algebra I} by van der Waerden, Springer. 

24 
\end{itemize} 

25 
*} 

14504  26 

14738  27 
axclass semiring \<subseteq> ab_semigroup_add, semigroup_mult 
28 
left_distrib: "(a + b) * c = a * c + b * c" 

29 
right_distrib: "a * (b + c) = a * b + a * c" 

14504  30 

14738  31 
axclass semiring_0 \<subseteq> semiring, comm_monoid_add 
14504  32 

14738  33 
axclass comm_semiring \<subseteq> ab_semigroup_add, ab_semigroup_mult 
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mult_commute: "a * b = b * a" 

35 
distrib: "(a + b) * c = a * c + b * c" 

14504  36 

14738  37 
instance comm_semiring \<subseteq> semiring 
38 
proof 

39 
fix a b c :: 'a 

40 
show "(a + b) * c = a * c + b * c" by (simp add: distrib) 

41 
have "a * (b + c) = (b + c) * a" by (simp add: mult_ac) 

42 
also have "... = b * a + c * a" by (simp only: distrib) 

43 
also have "... = a * b + a * c" by (simp add: mult_ac) 

44 
finally show "a * (b + c) = a * b + a * c" by blast 

14504  45 
qed 
46 

14738  47 
axclass comm_semiring_0 \<subseteq> comm_semiring, comm_monoid_add 
14504  48 

14738  49 
instance comm_semiring_0 \<subseteq> semiring_0 .. 
14504  50 

14738  51 
axclass axclass_0_neq_1 \<subseteq> zero, one 
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zero_neq_one [simp]: "0 \<noteq> 1" 
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14738  54 
axclass semiring_1 \<subseteq> axclass_0_neq_1, semiring_0, monoid_mult 
14504  55 

14738  56 
axclass comm_semiring_1 \<subseteq> axclass_0_neq_1, comm_semiring_0, comm_monoid_mult (* previously almost_semiring *) 
57 

58 
instance comm_semiring_1 \<subseteq> semiring_1 .. 

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14738  60 
axclass axclass_no_zero_divisors \<subseteq> zero, times 
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no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0" 

14504  62 

14738  63 
axclass comm_semiring_1_cancel \<subseteq> comm_semiring_1, cancel_ab_semigroup_add (* previously semiring *) 
64 

65 
axclass ring \<subseteq> semiring, ab_group_add 

66 

67 
instance ring \<subseteq> semiring_0 .. 

14504  68 

14738  69 
axclass comm_ring \<subseteq> comm_semiring_0, ab_group_add 
70 

71 
instance comm_ring \<subseteq> ring .. 

14504  72 

14738  73 
instance comm_ring \<subseteq> comm_semiring_0 .. 
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75 
axclass ring_1 \<subseteq> ring, semiring_1 

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axclass comm_ring_1 \<subseteq> comm_ring, comm_semiring_1 (* previously ring *) 
78 

79 
instance comm_ring_1 \<subseteq> ring_1 .. 

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14738  81 
instance comm_ring_1 \<subseteq> comm_semiring_1_cancel .. 
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14738  83 
axclass idom \<subseteq> comm_ring_1, axclass_no_zero_divisors 
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14738  85 
axclass field \<subseteq> comm_ring_1, inverse 
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left_inverse [simp]: "a \<noteq> 0 ==> inverse a * a = 1" 
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divide_inverse: "a / b = a * inverse b" 
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14738  89 
lemma mult_zero_left [simp]: "0 * a = (0::'a::{semiring_0, cancel_semigroup_add})" 
90 
proof  

91 
have "0*a + 0*a = 0*a + 0" 

92 
by (simp add: left_distrib [symmetric]) 

93 
thus ?thesis 

94 
by (simp only: add_left_cancel) 

95 
qed 

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14738  97 
lemma mult_zero_right [simp]: "a * 0 = (0::'a::{semiring_0, cancel_semigroup_add})" 
98 
proof  

99 
have "a*0 + a*0 = a*0 + 0" 

100 
by (simp add: right_distrib [symmetric]) 

101 
thus ?thesis 

102 
by (simp only: add_left_cancel) 

103 
qed 

104 

105 
lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0  b = 0)" 

106 
proof cases 

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assume "a=0" thus ?thesis by simp 

108 
next 

109 
assume anz [simp]: "a\<noteq>0" 

110 
{ assume "a * b = 0" 

111 
hence "inverse a * (a * b) = 0" by simp 

112 
hence "b = 0" by (simp (no_asm_use) add: mult_assoc [symmetric])} 

113 
thus ?thesis by force 

114 
qed 

115 

116 
instance field \<subseteq> idom 

117 
by (intro_classes, simp) 

118 

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axclass division_by_zero \<subseteq> zero, inverse 
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inverse_zero [simp]: "inverse 0 = 0" 
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subsection {* Distribution rules *} 
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theorems ring_distrib = right_distrib left_distrib 
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text{*For the @{text combine_numerals} simproc*} 
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lemma combine_common_factor: 
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"a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)" 
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by (simp add: left_distrib add_ac) 
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lemma minus_mult_left: " (a * b) = (a) * (b::'a::ring)" 
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apply (rule equals_zero_I) 
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apply (simp add: left_distrib [symmetric]) 
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done 
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lemma minus_mult_right: " (a * b) = a * (b::'a::ring)" 
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apply (rule equals_zero_I) 
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apply (simp add: right_distrib [symmetric]) 
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done 
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lemma minus_mult_minus [simp]: "( a) * ( b) = a * (b::'a::ring)" 
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by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemma minus_mult_commute: "( a) * b = a * ( b::'a::ring)" 
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by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemma right_diff_distrib: "a * (b  c) = a * b  a * (c::'a::ring)" 
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by (simp add: right_distrib diff_minus 
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minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemma left_diff_distrib: "(a  b) * c = a * c  b * (c::'a::ring)" 
14738  152 
by (simp add: left_distrib diff_minus 
153 
minus_mult_left [symmetric] minus_mult_right [symmetric]) 

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14738  155 
axclass pordered_semiring \<subseteq> semiring_0, pordered_ab_semigroup_add 
156 
mult_left_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b" 

157 
mult_right_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> a * c <= b * c" 

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14738  159 
axclass pordered_cancel_semiring \<subseteq> pordered_semiring, cancel_ab_semigroup_add 
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14738  161 
axclass ordered_semiring_strict \<subseteq> semiring_0, ordered_cancel_ab_semigroup_add 
162 
mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" 

163 
mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c" 

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14738  165 
instance ordered_semiring_strict \<subseteq> pordered_cancel_semiring 
166 
apply intro_classes 

167 
apply (case_tac "a < b & 0 < c") 

168 
apply (auto simp add: mult_strict_left_mono order_less_le) 

169 
apply (auto simp add: mult_strict_left_mono order_le_less) 

170 
apply (simp add: mult_strict_right_mono) 

14270  171 
done 
172 

14738  173 
axclass pordered_comm_semiring \<subseteq> comm_semiring_0, pordered_ab_semigroup_add 
174 
mult_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b" 

14270  175 

14738  176 
axclass pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring, cancel_ab_semigroup_add 
14270  177 

14738  178 
instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring .. 
14270  179 

14738  180 
axclass ordered_comm_semiring_strict \<subseteq> comm_semiring_0, ordered_cancel_ab_semigroup_add 
181 
mult_strict_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" 

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14738  183 
instance pordered_comm_semiring \<subseteq> pordered_semiring 
184 
by (intro_classes, insert mult_mono, simp_all add: mult_commute, blast+) 

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14738  186 
instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring .. 
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14738  188 
instance ordered_comm_semiring_strict \<subseteq> ordered_semiring_strict 
189 
by (intro_classes, insert mult_strict_mono, simp_all add: mult_commute, blast+) 

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14738  191 
instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring 
192 
apply (intro_classes) 

193 
apply (case_tac "a < b & 0 < c") 

194 
apply (auto simp add: mult_strict_left_mono order_less_le) 

195 
apply (auto simp add: mult_strict_left_mono order_le_less) 

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done 
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14738  198 
axclass pordered_ring \<subseteq> ring, pordered_semiring 
14270  199 

14738  200 
instance pordered_ring \<subseteq> pordered_ab_group_add .. 
14270  201 

14738  202 
instance pordered_ring \<subseteq> pordered_cancel_semiring .. 
14270  203 

14738  204 
axclass lordered_ring \<subseteq> pordered_ring, lordered_ab_group_abs 
14270  205 

14738  206 
axclass axclass_abs_if \<subseteq> minus, ord, zero 
207 
abs_if: "abs a = (if (a < 0) then (a) else a)" 

14270  208 

14738  209 
axclass ordered_ring_strict \<subseteq> ring, ordered_semiring_strict, axclass_abs_if 
14270  210 

14738  211 
instance ordered_ring_strict \<subseteq> lordered_ab_group .. 
14270  212 

14738  213 
instance ordered_ring_strict \<subseteq> lordered_ring 
214 
by (intro_classes, simp add: abs_if join_eq_if) 

14270  215 

14738  216 
axclass pordered_comm_ring \<subseteq> comm_ring, pordered_comm_semiring 
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14738  218 
axclass ordered_semidom \<subseteq> comm_semiring_1_cancel, ordered_comm_semiring_strict (* previously ordered_semiring *) 
219 
zero_less_one [simp]: "0 < 1" 

14270  220 

14738  221 
axclass ordered_idom \<subseteq> comm_ring_1, ordered_comm_semiring_strict, axclass_abs_if (* previously ordered_ring *) 
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14738  223 
instance ordered_idom \<subseteq> ordered_ring_strict .. 
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14738  225 
axclass ordered_field \<subseteq> field, ordered_idom 
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lemma eq_add_iff1: 
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"(a*e + c = b*e + d) = ((ab)*e + c = (d::'a::ring))" 
14738  229 
apply (simp add: diff_minus left_distrib) 
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apply (simp add: diff_minus left_distrib add_ac) 
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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done 
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lemma eq_add_iff2: 
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"(a*e + c = b*e + d) = (c = (ba)*e + (d::'a::ring))" 
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apply (simp add: diff_minus left_distrib add_ac) 
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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done 
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lemma less_add_iff1: 
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"(a*e + c < b*e + d) = ((ab)*e + c < (d::'a::pordered_ring))" 
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apply (simp add: diff_minus left_distrib add_ac) 
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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done 
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lemma less_add_iff2: 
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"(a*e + c < b*e + d) = (c < (ba)*e + (d::'a::pordered_ring))" 
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apply (simp add: diff_minus left_distrib add_ac) 
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Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

249 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

250 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

251 

5efbb548107d
Tidying of the integer development; towards removing the
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parents:
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diff
changeset

252 
lemma le_add_iff1: 
14738  253 
"(a*e + c \<le> b*e + d) = ((ab)*e + c \<le> (d::'a::pordered_ring))" 
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

254 
apply (simp add: diff_minus left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

255 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

256 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

257 

5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

258 
lemma le_add_iff2: 
14738  259 
"(a*e + c \<le> b*e + d) = (c \<le> (ba)*e + (d::'a::pordered_ring))" 
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

260 
apply (simp add: diff_minus left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

261 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

262 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

263 

14270  264 
subsection {* Ordering Rules for Multiplication *} 
265 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
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14341
diff
changeset

266 
lemma mult_left_le_imp_le: 
14738  267 
"[c*a \<le> c*b; 0 < c] ==> a \<le> (b::'a::ordered_semiring_strict)" 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

268 
by (force simp add: mult_strict_left_mono linorder_not_less [symmetric]) 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

269 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

270 
lemma mult_right_le_imp_le: 
14738  271 
"[a*c \<le> b*c; 0 < c] ==> a \<le> (b::'a::ordered_semiring_strict)" 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

272 
by (force simp add: mult_strict_right_mono linorder_not_less [symmetric]) 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

273 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

274 
lemma mult_left_less_imp_less: 
14738  275 
"[c*a < c*b; 0 \<le> c] ==> a < (b::'a::ordered_semiring_strict)" 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

276 
by (force simp add: mult_left_mono linorder_not_le [symmetric]) 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

277 

744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

278 
lemma mult_right_less_imp_less: 
14738  279 
"[a*c < b*c; 0 \<le> c] ==> a < (b::'a::ordered_semiring_strict)" 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

280 
by (force simp add: mult_right_mono linorder_not_le [symmetric]) 
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

281 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

282 
lemma mult_strict_left_mono_neg: 
14738  283 
"[b < a; c < 0] ==> c * a < c * (b::'a::ordered_ring_strict)" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

284 
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

285 
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

286 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

287 

14738  288 
lemma mult_left_mono_neg: 
289 
"[b \<le> a; c \<le> 0] ==> c * a \<le> c * (b::'a::pordered_ring)" 

290 
apply (drule mult_left_mono [of _ _ "c"]) 

291 
apply (simp_all add: minus_mult_left [symmetric]) 

292 
done 

293 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

294 
lemma mult_strict_right_mono_neg: 
14738  295 
"[b < a; c < 0] ==> a * c < b * (c::'a::ordered_ring_strict)" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

296 
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

297 
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

298 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

299 

14738  300 
lemma mult_right_mono_neg: 
301 
"[b \<le> a; c \<le> 0] ==> a * c \<le> (b::'a::pordered_ring) * c" 

302 
apply (drule mult_right_mono [of _ _ "c"]) 

303 
apply (simp) 

304 
apply (simp_all add: minus_mult_right [symmetric]) 

305 
done 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

306 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

307 
subsection{* Products of Signs *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

308 

14738  309 
lemma mult_pos: "[ (0::'a::ordered_semiring_strict) < 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

310 
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

311 

14738  312 
lemma mult_pos_le: "[ (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b ] ==> 0 \<le> a*b" 
313 
by (drule mult_left_mono [of 0 b], auto) 

314 

315 
lemma mult_pos_neg: "[ (0::'a::ordered_semiring_strict) < 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

316 
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

317 

14738  318 
lemma mult_pos_neg_le: "[ (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 ] ==> a*b \<le> 0" 
319 
by (drule mult_left_mono [of b 0], auto) 

320 

321 
lemma mult_pos_neg2: "[ (0::'a::ordered_semiring_strict) < a; b < 0 ] ==> b*a < 0" 

322 
by (drule mult_strict_right_mono[of b 0], auto) 

323 

324 
lemma mult_pos_neg2_le: "[ (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 ] ==> b*a \<le> 0" 

325 
by (drule mult_right_mono[of b 0], auto) 

326 

327 
lemma mult_neg: "[ a < (0::'a::ordered_ring_strict); b < 0 ] ==> 0 < a*b" 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

328 
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

329 

14738  330 
lemma mult_neg_le: "[ a \<le> (0::'a::pordered_ring); b \<le> 0 ] ==> 0 \<le> a*b" 
331 
by (drule mult_right_mono_neg[of a 0 b ], auto) 

332 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

333 
lemma zero_less_mult_pos: 
14738  334 
"[ 0 < a*b; 0 < a] ==> 0 < (b::'a::ordered_semiring_strict)" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

335 
apply (case_tac "b\<le>0") 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

336 
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

337 
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

338 
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

339 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

340 

14738  341 
lemma zero_less_mult_pos2: 
342 
"[ 0 < b*a; 0 < a] ==> 0 < (b::'a::ordered_semiring_strict)" 

343 
apply (case_tac "b\<le>0") 

344 
apply (auto simp add: order_le_less linorder_not_less) 

345 
apply (drule_tac mult_pos_neg2 [of a b]) 

346 
apply (auto dest: order_less_not_sym) 

347 
done 

348 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

349 
lemma zero_less_mult_iff: 
14738  350 
"((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b  a < 0 & b < 0)" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

351 
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

352 
apply (blast dest: zero_less_mult_pos) 
14738  353 
apply (blast dest: zero_less_mult_pos2) 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

354 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

355 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14334
diff
changeset

356 
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

357 
assumption of an ordering. See @{text field_mult_eq_0_iff} below.*} 
14738  358 
lemma mult_eq_0_iff [simp]: "(a*b = (0::'a::ordered_ring_strict)) = (a = 0  b = 0)" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

359 
apply (case_tac "a < 0") 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

360 
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

361 
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

362 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

363 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

364 
lemma zero_le_mult_iff: 
14738  365 
"((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b  a \<le> 0 & b \<le> 0)" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

366 
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

367 
zero_less_mult_iff) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

368 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

369 
lemma mult_less_0_iff: 
14738  370 
"(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0  a < 0 & 0 < b)" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

371 
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

372 
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

373 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

374 

95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

375 
lemma mult_le_0_iff: 
14738  376 
"(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0  a \<le> 0 & 0 \<le> b)" 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

377 
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

378 
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

379 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

380 

14738  381 
lemma split_mult_pos_le: "(0 \<le> a & 0 \<le> b)  (a \<le> 0 & b \<le> 0) \<Longrightarrow> 0 \<le> a * (b::_::pordered_ring)" 
382 
by (auto simp add: mult_pos_le mult_neg_le) 

383 

384 
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0)  (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> (0::_::pordered_cancel_semiring)" 

385 
by (auto simp add: mult_pos_neg_le mult_pos_neg2_le) 

386 

387 
lemma zero_le_square: "(0::'a::ordered_ring_strict) \<le> a*a" 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

388 
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

389 

14738  390 
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom} 
391 
theorems available to members of @{term ordered_idom} *} 

392 

393 
instance ordered_idom \<subseteq> ordered_semidom 

14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset

394 
proof 
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset

395 
have "(0::'a) \<le> 1*1" by (rule zero_le_square) 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

396 
thus "(0::'a) < 1" by (simp add: order_le_less) 
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset

397 
qed 
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset

398 

14738  399 
instance ordered_ring_strict \<subseteq> axclass_no_zero_divisors 
400 
by (intro_classes, simp) 

401 

402 
instance ordered_idom \<subseteq> idom .. 

403 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

404 
text{*All three types of comparision involving 0 and 1 are covered.*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

405 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

406 
declare zero_neq_one [THEN not_sym, simp] 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

407 

14738  408 
lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

409 
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

410 

14738  411 
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0" 
412 
by (simp add: linorder_not_le) 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

413 

14738  414 
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0" 
415 
by (simp add: linorder_not_less) 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

416 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

417 
subsection{*More Monotonicity*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

418 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

419 
lemma mult_left_mono_neg: 
14738  420 
"[b \<le> a; c \<le> 0] ==> c * a \<le> c * (b::'a::pordered_ring)" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

421 
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

422 
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

423 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

424 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

425 
lemma mult_right_mono_neg: 
14738  426 
"[b \<le> a; c \<le> 0] ==> a * c \<le> b * (c::'a::pordered_ring)" 
427 
apply (drule mult_right_mono [of _ _ "c"]) 

428 
apply (simp_all add: minus_mult_right [symmetric]) 

429 
done 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

430 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

431 
text{*Strict monotonicity in both arguments*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

432 
lemma mult_strict_mono: 
14738  433 
"[a<b; c<d; 0<b; 0\<le>c] ==> a * c < b * (d::'a::ordered_semiring_strict)" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

434 
apply (case_tac "c=0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

435 
apply (simp add: mult_pos) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

436 
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

437 
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

438 
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

439 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

440 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

441 
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

442 
lemma mult_strict_mono': 
14738  443 
"[ a<b; c<d; 0 \<le> a; 0 \<le> c] ==> a * c < b * (d::'a::ordered_semiring_strict)" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

444 
apply (rule mult_strict_mono) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

445 
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

446 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

447 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

448 
lemma mult_mono: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

449 
"[a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c] 
14738  450 
==> a * c \<le> b * (d::'a::pordered_semiring)" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

451 
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

452 
apply (erule mult_left_mono, assumption) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

453 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

454 

14738  455 
lemma less_1_mult: "[ 1 < m; 1 < n ] ==> 1 < m*(n::'a::ordered_semidom)" 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

456 
apply (insert mult_strict_mono [of 1 m 1 n]) 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

457 
apply (simp add: order_less_trans [OF zero_less_one]) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

458 
done 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

459 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

460 
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

461 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

462 
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

463 
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

464 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

465 
lemma mult_less_cancel_right: 
14738  466 
"(a*c < b*c) = ((0 < c & a < b)  (c < 0 & b < (a::'a::ordered_ring_strict)))" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

467 
apply (case_tac "c = 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

468 
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

469 
mult_strict_right_mono_neg) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

470 
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

471 
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

472 
linorder_not_le [symmetric, of a]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

473 
apply (erule_tac [!] notE) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

474 
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

475 
mult_right_mono_neg) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

476 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

477 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

478 
lemma mult_less_cancel_left: 
14738  479 
"(c*a < c*b) = ((0 < c & a < b)  (c < 0 & b < (a::'a::ordered_ring_strict)))" 
480 
apply (case_tac "c = 0") 

481 
apply (auto simp add: linorder_neq_iff mult_strict_left_mono 

482 
mult_strict_left_mono_neg) 

483 
apply (auto simp add: linorder_not_less 

484 
linorder_not_le [symmetric, of "c*a"] 

485 
linorder_not_le [symmetric, of a]) 

486 
apply (erule_tac [!] notE) 

487 
apply (auto simp add: order_less_imp_le mult_left_mono 

488 
mult_left_mono_neg) 

489 
done 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

490 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

491 
lemma mult_le_cancel_right: 
14738  492 
"(a*c \<le> b*c) = ((0<c > a\<le>b) & (c<0 > b \<le> (a::'a::ordered_ring_strict)))" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

493 
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

494 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

495 
lemma mult_le_cancel_left: 
14738  496 
"(c*a \<le> c*b) = ((0<c > a\<le>b) & (c<0 > b \<le> (a::'a::ordered_ring_strict)))" 
497 
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left) 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

498 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

499 
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

500 
assumes less: "c*a < c*b" and nonneg: "0 \<le> c" 
14738  501 
shows "a < (b::'a::ordered_semiring_strict)" 
14377  502 
proof (rule ccontr) 
503 
assume "~ a < b" 

504 
hence "b \<le> a" by (simp add: linorder_not_less) 

505 
hence "c*b \<le> c*a" by (rule mult_left_mono) 

506 
with this and less show False 

507 
by (simp add: linorder_not_less [symmetric]) 

508 
qed 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

509 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

510 
lemma mult_less_imp_less_right: 
14738  511 
assumes less: "a*c < b*c" and nonneg: "0 <= c" 
512 
shows "a < (b::'a::ordered_semiring_strict)" 

513 
proof (rule ccontr) 

514 
assume "~ a < b" 

515 
hence "b \<le> a" by (simp add: linorder_not_less) 

516 
hence "b*c \<le> a*c" by (rule mult_right_mono) 

517 
with this and less show False 

518 
by (simp add: linorder_not_less [symmetric]) 

519 
qed 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

520 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

521 
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

522 
lemma mult_cancel_right [simp]: 
14738  523 
"(a*c = b*c) = (c = (0::'a::ordered_ring_strict)  a=b)" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

524 
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

525 
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

526 
simp add: linorder_neq_iff) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

527 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

528 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

529 
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

530 
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

531 
lemma mult_cancel_left [simp]: 
14738  532 
"(c*a = c*b) = (c = (0::'a::ordered_ring_strict)  a=b)" 
533 
apply (cut_tac linorder_less_linear [of 0 c]) 

534 
apply (force dest: mult_strict_left_mono_neg mult_strict_left_mono 

535 
simp add: linorder_neq_iff) 

536 
done 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

537 

14738  538 
text{*This list of rewrites decides ring equalities by ordered rewriting.*} 
539 
lemmas ring_eq_simps = 

540 
mult_ac 

541 
left_distrib right_distrib left_diff_distrib right_diff_distrib 

542 
add_ac 

543 
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 

544 
diff_eq_eq eq_diff_eq 

545 

546 
thm ring_eq_simps 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

547 
subsection {* Fields *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

548 

14288  549 
lemma right_inverse [simp]: 
550 
assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1" 

551 
proof  

552 
have "a * inverse a = inverse a * a" by (simp add: mult_ac) 

553 
also have "... = 1" using not0 by simp 

554 
finally show ?thesis . 

555 
qed 

556 

557 
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))" 

558 
proof 

559 
assume neq: "b \<noteq> 0" 

560 
{ 

561 
hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac) 

562 
also assume "a / b = 1" 

563 
finally show "a = b" by simp 

564 
next 

565 
assume "a = b" 

566 
with neq show "a / b = 1" by (simp add: divide_inverse) 

567 
} 

568 
qed 

569 

570 
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a" 

571 
by (simp add: divide_inverse) 

572 

573 
lemma divide_self [simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1" 

574 
by (simp add: divide_inverse) 

575 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

576 
lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

577 
by (simp add: divide_inverse) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

578 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

579 
lemma divide_zero_left [simp]: "0/a = (0::'a::field)" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

580 
by (simp add: divide_inverse) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

581 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

582 
lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

583 
by (simp add: divide_inverse) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

584 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

585 
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c" 
14293  586 
by (simp add: divide_inverse left_distrib) 
587 

588 

14270  589 
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement 
590 
of an ordering.*} 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

591 
lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0  b = 0)" 
14377  592 
proof cases 
593 
assume "a=0" thus ?thesis by simp 

594 
next 

595 
assume anz [simp]: "a\<noteq>0" 

596 
{ assume "a * b = 0" 

597 
hence "inverse a * (a * b) = 0" by simp 

598 
hence "b = 0" by (simp (no_asm_use) add: mult_assoc [symmetric])} 

599 
thus ?thesis by force 

600 
qed 

14270  601 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

602 
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

603 
lemma field_mult_cancel_right_lemma: 
14269  604 
assumes cnz: "c \<noteq> (0::'a::field)" 
605 
and eq: "a*c = b*c" 

606 
shows "a=b" 

14377  607 
proof  
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

608 
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

609 
by (simp add: eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

610 
thus "a=b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

611 
by (simp add: mult_assoc cnz) 
14377  612 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

613 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

614 
lemma field_mult_cancel_right [simp]: 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

615 
"(a*c = b*c) = (c = (0::'a::field)  a=b)" 
14377  616 
proof cases 
617 
assume "c=0" thus ?thesis by simp 

618 
next 

619 
assume "c\<noteq>0" 

620 
thus ?thesis by (force dest: field_mult_cancel_right_lemma) 

621 
qed 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

622 

14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

623 
lemma field_mult_cancel_left [simp]: 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

624 
"(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

625 
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

626 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

627 
lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)" 
14377  628 
proof 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

629 
assume ianz: "inverse a = 0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

630 
assume "a \<noteq> 0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

631 
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

632 
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

633 
finally have "1 = (0::'a::field)" . 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

634 
thus False by (simp add: eq_commute) 
14377  635 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

636 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

637 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

638 
subsection{*Basic Properties of @{term inverse}*} 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

639 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

640 
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

641 
apply (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

642 
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

643 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

644 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

645 
lemma inverse_nonzero_imp_nonzero: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

646 
"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

647 
apply (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

648 
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

649 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

650 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

651 
lemma inverse_nonzero_iff_nonzero [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

652 
"(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

653 
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

654 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

655 
lemma nonzero_inverse_minus_eq: 
14269  656 
assumes [simp]: "a\<noteq>0" shows "inverse(a) = inverse(a::'a::field)" 
14377  657 
proof  
658 
have "a * inverse ( a) = a *  inverse a" 

659 
by simp 

660 
thus ?thesis 

661 
by (simp only: field_mult_cancel_left, simp) 

662 
qed 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

663 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

664 
lemma inverse_minus_eq [simp]: 
14377  665 
"inverse(a) = inverse(a::'a::{field,division_by_zero})"; 
666 
proof cases 

667 
assume "a=0" thus ?thesis by (simp add: inverse_zero) 

668 
next 

669 
assume "a\<noteq>0" 

670 
thus ?thesis by (simp add: nonzero_inverse_minus_eq) 

671 
qed 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

672 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

673 
lemma nonzero_inverse_eq_imp_eq: 
14269  674 
assumes inveq: "inverse a = inverse b" 
675 
and anz: "a \<noteq> 0" 

676 
and bnz: "b \<noteq> 0" 

677 
shows "a = (b::'a::field)" 

14377  678 
proof  
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

679 
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

680 
by (simp add: inveq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

681 
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

682 
by simp 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

683 
thus "a = b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

684 
by (simp add: mult_assoc anz bnz) 
14377  685 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

686 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

687 
lemma inverse_eq_imp_eq: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

688 
"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

689 
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

690 
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

691 
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

692 
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

693 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

694 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

695 
lemma inverse_eq_iff_eq [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

696 
"(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

697 
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

698 

14270  699 
lemma nonzero_inverse_inverse_eq: 
700 
assumes [simp]: "a \<noteq> 0" shows "inverse(inverse (a::'a::field)) = a" 

701 
proof  

702 
have "(inverse (inverse a) * inverse a) * a = a" 

703 
by (simp add: nonzero_imp_inverse_nonzero) 

704 
thus ?thesis 

705 
by (simp add: mult_assoc) 

706 
qed 

707 

708 
lemma inverse_inverse_eq [simp]: 

709 
"inverse(inverse (a::'a::{field,division_by_zero})) = a" 

710 
proof cases 

711 
assume "a=0" thus ?thesis by simp 

712 
next 

713 
assume "a\<noteq>0" 

714 
thus ?thesis by (simp add: nonzero_inverse_inverse_eq) 

715 
qed 

716 

717 
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)" 

718 
proof  

719 
have "inverse 1 * 1 = (1::'a::field)" 

720 
by (rule left_inverse [OF zero_neq_one [symmetric]]) 

721 
thus ?thesis by simp 

722 
qed 

723 

724 
lemma nonzero_inverse_mult_distrib: 

725 
assumes anz: "a \<noteq> 0" 

726 
and bnz: "b \<noteq> 0" 

727 
shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)" 

728 
proof  

729 
have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 

730 
by (simp add: field_mult_eq_0_iff anz bnz) 

731 
hence "inverse(a*b) * a = inverse(b)" 

732 
by (simp add: mult_assoc bnz) 

733 
hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 

734 
by simp 

735 
thus ?thesis 

736 
by (simp add: mult_assoc anz) 

737 
qed 

738 

739 
text{*This version builds in division by zero while also reorienting 

740 
the righthand side.*} 

741 
lemma inverse_mult_distrib [simp]: 

742 
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})" 

743 
proof cases 

744 
assume "a \<noteq> 0 & b \<noteq> 0" 

745 
thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_commute) 

746 
next 

747 
assume "~ (a \<noteq> 0 & b \<noteq> 0)" 

748 
thus ?thesis by force 

749 
qed 

750 

751 
text{*There is no slick version using division by zero.*} 

752 
lemma inverse_add: 

753 
"[a \<noteq> 0; b \<noteq> 0] 

754 
==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)" 

755 
apply (simp add: left_distrib mult_assoc) 

756 
apply (simp add: mult_commute [of "inverse a"]) 

757 
apply (simp add: mult_assoc [symmetric] add_commute) 

758 
done 

759 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

760 
lemma inverse_divide [simp]: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

761 
"inverse (a/b) = b / (a::'a::{field,division_by_zero})" 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

762 
by (simp add: divide_inverse mult_commute) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

763 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

764 
lemma nonzero_mult_divide_cancel_left: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

765 
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

766 
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

767 
proof  
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

768 
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

769 
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

770 
nonzero_inverse_mult_distrib) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

771 
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

772 
by (simp only: mult_ac) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

773 
also have "... = a * inverse b" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

774 
by simp 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

775 
finally show ?thesis 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

776 
by (simp add: divide_inverse) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

777 
qed 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

778 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

779 
lemma mult_divide_cancel_left: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

780 
"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

781 
apply (case_tac "b = 0") 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

782 
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

783 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

784 

14321  785 
lemma nonzero_mult_divide_cancel_right: 
786 
"[b\<noteq>0; c\<noteq>0] ==> (a*c) / (b*c) = a/(b::'a::field)" 

787 
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 

788 

789 
lemma mult_divide_cancel_right: 

790 
"c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})" 

791 
apply (case_tac "b = 0") 

792 
apply (simp_all add: nonzero_mult_divide_cancel_right) 

793 
done 

794 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

795 
(*For ExtractCommonTerm*) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

796 
lemma mult_divide_cancel_eq_if: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

797 
"(c*a) / (c*b) = 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

798 
(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

799 
by (simp add: mult_divide_cancel_left) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

800 

14284
f1abe67c448a
reorganisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset

801 
lemma divide_1 [simp]: "a/1 = (a::'a::field)" 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

802 
by (simp add: divide_inverse) 
14284
f1abe67c448a
reorganisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset

803 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

804 
lemma times_divide_eq_right [simp]: "a * (b/c) = (a*b) / (c::'a::field)" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

805 
by (simp add: divide_inverse mult_assoc) 
14288  806 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

807 
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

808 
by (simp add: divide_inverse mult_ac) 
14288  809 

810 
lemma divide_divide_eq_right [simp]: 

811 
"a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})" 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

812 
by (simp add: divide_inverse mult_ac) 
14288  813 

814 
lemma divide_divide_eq_left [simp]: 

815 
"(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)" 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

816 
by (simp add: divide_inverse mult_assoc) 
14288  817 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

818 

14293  819 
subsection {* Division and Unary Minus *} 
820 

821 
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==>  (a/b) = (a) / (b::'a::field)" 

822 
by (simp add: divide_inverse minus_mult_left) 

823 

824 
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==>  (a/b) = a / (b::'a::field)" 

825 
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right) 

826 

827 
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (a)/(b) = a / (b::'a::field)" 

828 
by (simp add: divide_inverse nonzero_inverse_minus_eq) 

829 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

830 
lemma minus_divide_left: " (a/b) = (a) / (b::'a::field)" 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

831 
by (simp add: divide_inverse minus_mult_left [symmetric]) 
14293  832 

833 
lemma minus_divide_right: " (a/b) = a / (b::'a::{field,division_by_zero})" 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

834 
by (simp add: divide_inverse minus_mult_right [symmetric]) 
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

835 

14293  836 

837 
text{*The effect is to extract signs from divisions*} 

838 
declare minus_divide_left [symmetric, simp] 

839 
declare minus_divide_right [symmetric, simp] 

840 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

841 
text{*Also, extract signs from products*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

842 
declare minus_mult_left [symmetric, simp] 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

843 
declare minus_mult_right [symmetric, simp] 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

844 

14293  845 
lemma minus_divide_divide [simp]: 
846 
"(a)/(b) = a / (b::'a::{field,division_by_zero})" 

847 
apply (case_tac "b=0", simp) 

848 
apply (simp add: nonzero_minus_divide_divide) 

849 
done 

850 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

851 
lemma diff_divide_distrib: "(ab)/(c::'a::field) = a/c  b/c" 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

852 
by (simp add: diff_minus add_divide_distrib) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

853 

14293  854 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

855 
subsection {* Ordered Fields *} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

856 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

857 
lemma positive_imp_inverse_positive: 
14269  858 
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

859 
proof  
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

860 
have "0 < a * inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

861 
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

862 
thus "0 < inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

863 
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

864 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

865 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

866 
lemma negative_imp_inverse_negative: 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

867 
"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

868 
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

869 
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

870 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

871 
lemma inverse_le_imp_le: 
14269  872 
assumes invle: "inverse a \<le> inverse b" 
873 
and apos: "0 < a" 

874 
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

875 
proof (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

876 
assume "~ b \<le> a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

877 
hence "a < b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

878 
by (simp add: linorder_not_le) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

879 
hence bpos: "0 < b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

880 
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

881 
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

882 
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

883 
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

884 
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

885 
thus "b \<le> a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

886 
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

887 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

888 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

889 
lemma inverse_positive_imp_positive: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

890 
assumes inv_gt_0: "0 < inverse a" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

891 
and [simp]: "a \<noteq> 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

892 
shows "0 < (a::'a::ordered_field)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

893 
proof  
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

894 
have "0 < inverse (inverse a)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

895 
by (rule positive_imp_inverse_positive) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

896 
thus "0 < a" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

897 
by (simp add: nonzero_inverse_inverse_eq) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

898 
qed 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

899 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

900 
lemma inverse_positive_iff_positive [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

901 
"(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

902 
apply (case_tac "a = 0", simp) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

903 
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

904 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

905 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

906 
lemma inverse_negative_imp_negative: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

907 
assumes inv_less_0: "inverse a < 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

908 
and [simp]: "a \<noteq> 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

909 
shows "a < (0::'a::ordered_field)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

910 
proof  
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

911 
have "inverse (inverse a) < 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

912 
by (rule negative_imp_inverse_negative) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

913 
thus "a < 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

914 
by (simp add: nonzero_inverse_inverse_eq) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

915 
qed 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

916 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

917 
lemma inverse_negative_iff_negative [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

918 
"(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

919 
apply (case_tac "a = 0", simp) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

920 
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

921 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

922 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

923 
lemma inverse_nonnegative_iff_nonnegative [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

924 
"(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

925 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

926 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

927 
lemma inverse_nonpositive_iff_nonpositive [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

928 
"(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

929 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

930 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

931 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

932 
subsection{*AntiMonotonicity of @{term inverse}*} 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

933 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

934 
lemma less_imp_inverse_less: 
14269  935 
assumes less: "a < b" 
936 
and apos: "0 < a" 

937 
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

938 
proof (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

939 
assume "~ inverse b < inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

940 
hence "inverse a \<le> inverse b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

941 
by (simp add: linorder_not_less) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

942 
hence "~ (a < b)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

943 
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

944 
thus False 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

945 
by (rule notE [OF _ less]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

946 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

947 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

948 
lemma inverse_less_imp_less: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

949 
"[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

950 
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

951 
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

952 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

953 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

954 
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

955 
lemma inverse_less_iff_less [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

956 
"[0 < a; 0 < b] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

957 
==> (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

958 
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

959 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

960 
lemma le_imp_inverse_le: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

961 
"[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

962 
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

963 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

964 
lemma inverse_le_iff_le [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

965 
"[0 < a; 0 < b] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

966 
==> (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

967 
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

968 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

969 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

970 
text{*These results refer to both operands being negative. The oppositesign 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

971 
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

972 
lemma inverse_le_imp_le_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

973 
"[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

974 
apply (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

975 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

976 
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

977 
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

978 
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

979 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

980 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

981 
lemma less_imp_inverse_less_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

982 
"[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

983 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

984 
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

985 
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

986 
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

987 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

988 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

989 
lemma inverse_less_imp_less_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

990 
"[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

991 
apply (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

992 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

993 
prefer 2 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

994 
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

995 
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

996 
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

997 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

998 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

999 
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

1000 
"[a < 0; b < 0] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1001 
==> (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

1002 
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

1003 
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

1004 
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

1005 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1006 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1007 
lemma le_imp_inverse_le_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1008 
"[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

1009 
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

1010 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1011 
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

1012 
"[a < 0; b < 0] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1013 
==> (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

1014 
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

1015 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1016 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1017 
subsection{*Inverses and the Number One*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1018 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1019 
lemma one_less_inverse_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1020 
"(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1021 
assume "0 < x" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1022 
with inverse_less_iff_less [OF zero_less_one, of x] 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1023 
show ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1024 
next 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1025 
assume notless: "~ (0 < x)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1026 
have "~ (1 < inverse x)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1027 
proof 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1028 
assume "1 < inverse x" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1029 
also with notless have "... \<le> 0" by (simp add: linorder_not_less) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1030 
also have "... < 1" by (rule zero_less_one) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1031 
finally show False by auto 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1032 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1033 
with notless show ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1034 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1035 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1036 
lemma inverse_eq_1_iff [simp]: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1037 
"(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1038 
by (insert inverse_eq_iff_eq [of x 1], simp) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1039 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1040 
lemma one_le_inverse_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1041 
"(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1042 
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1043 
eq_commute [of 1]) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1044 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1045 
lemma inverse_less_1_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1046 
"(inverse x < 1) = (x \<le> 0  1 < (x::'a::{ordered_field,division_by_zero}))" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1047 
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1048 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1049 
lemma inverse_le_1_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1050 
"(inverse x \<le> 1) = (x \<le> 0  1 \<le> (x::'a::{ordered_field,division_by_zero}))" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1051 
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1052 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1053 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1054 
subsection{*Division and Signs*} 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1055 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1056 
lemma zero_less_divide_iff: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1057 
"((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b  a < 0 & b < 0)" 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

1058 
by (simp add: divide_inverse zero_less_mult_iff) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1059 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1060 
lemma divide_less_0_iff: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1061 
"(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

1062 
(0 < a & b < 0  a < 0 & 0 < b)" 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

1063 
by (simp add: divide_inverse mult_less_0_iff) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1064 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1065 
lemma zero_le_divide_iff: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1066 
"((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

1067 
(0 \<le> a & 0 \<le> b  a \<le> 0 & b \<le> 0)" 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

1068 
by (simp add: divide_inverse zero_le_mult_iff) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1069 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1070 
lemma divide_le_0_iff: 
14288  1071 
"(a/b \<le> (0::'a::{ordered_field,division_by_zero})) = 
1072 
(0 \<le> a & b \<le> 0  a \<le> 0 & 0 \<le> b)" 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

1073 
by (simp add: divide_inverse mult_le_0_iff) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1074 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1075 
lemma divide_eq_0_iff [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1076 
"(a/b = 0) = (a=0  b=(0::'a::{field,division_by_zero}))" 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

1077 
by (simp add: divide_inverse field_mult_eq_0_iff) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1078 

14288  1079 

1080 
subsection{*Simplification of Inequalities Involving Literal Divisors*} 

1081 

1082 
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)" 

1083 
proof  

1084 
assume less: "0<c" 

1085 
hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)" 

1086 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

1087 
also have "... = (a*c \<le> b)" 

1088 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

1089 
finally show ?thesis . 

1090 
qed 

1091 

1092 

1093 
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)" 

1094 
proof  

1095 
assume less: "c<0" 

1096 
hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)" 

1097 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

1098 
also have "... = (b \<le> a*c)" 

1099 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

1100 
finally show ?thesis . 

1101 
qed 

1102 

1103 
lemma le_divide_eq: 

1104 
"(a \<le> b/c) = 

1105 
(if 0 < c then a*c \<le> b 

1106 
else if c < 0 then b \<le> a*c 

1107 
else a \<le> (0::'a::{ordered_field,division_by_zero}))" 

1108 
apply (case_tac "c=0", simp) 

1109 
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 

1110 
done 

1111 

1112 
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)" 

1113 
proof  

1114 
assume less: "0<c" 

1115 
hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)" 

1116 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

1117 
also have "... = (b \<le> a*c)" 

1118 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

1119 
finally show ?thesis . 

1120 
qed 

1121 

1122 
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)" 

1123 
proof  

1124 
assume less: "c<0" 

1125 
hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)" 

1126 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

1127 
also have "... = (a*c \<le> b)" 

1128 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

1129 
finally show ?thesis . 

1130 
qed 

1131 

1132 
lemma divide_le_eq: 

1133 
"(b/c \<le> a) = 

1134 
(if 0 < c then b \<le> a*c 

1135 
else if c < 0 then a*c \<le> b 

1136 
else 0 \<le> (a::'a::{ordered_field,division_by_zero}))" 

1137 
apply (case_tac "c=0", simp) 

1138 
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 

1139 
done 

1140 

1141 

1142 
lemma pos_less_divide_eq: 

1143 
"0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)" 

1144 
proof  

1145 
assume less: "0<c" 

1146 
hence "(a < b/c) = (a*c < (b/c)*c)" 

1147 
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) 

1148 
also have "... = (a*c < b)" 

1149 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

1150 
finally show ?thesis . 

1151 
qed 

1152 

1153 
lemma neg_less_divide_eq: 

1154 
"c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)" 

1155 
proof  

1156 
assume less: "c<0" 

1157 
hence "(a < b/c) = ((b/c)*c < a*c)" 

1158 
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) 

1159 
also have "... = (b < a*c)" 

1160 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

1161 
finally show ?thesis . 

1162 
qed 

1163 

1164 
lemma less_divide_eq: 

1165 
"(a < b/c) = 

1166 
(if 0 < c then a*c < b 

1167 
else if c < 0 then b < a*c 

1168 
else a < (0::'a::{ordered_field,division_by_zero}))" 

1169 
apply (case_tac "c=0", simp) 

1170 
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 

1171 
done 

1172 

1173 
lemma pos_divide_less_eq: 

1174 
"0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)" 

1175 
proof  

1176 
assume less: "0<c" 

1177 
hence "(b/c < a) = ((b/c)*c < a*c)" 

1178 
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) 

1179 
also have "... = (b < a*c)" 

1180 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

1181 
finally show ?thesis . 

1182 
qed 

1183 

1184 
lemma neg_divide_less_eq: 

1185 
"c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)" 

1186 
proof  

1187 
assume less: "c<0" 

1188 
hence "(b/c < a) = (a*c < (b/c)*c)" 

1189 
by (simp add: mult_less_cancel_right order_less_not_sym [OF less]) 

1190 
also have "... = (a*c < b)" 

1191 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

1192 
finally show ?thesis . 

1193 
qed 

1194 

1195 
lemma divide_less_eq: 

1196 
"(b/c < a) = 

1197 
(if 0 < c then b < a*c 

1198 
else if c < 0 then a*c < b 

1199 
else 0 < (a::'a::{ordered_field,division_by_zero}))" 

1200 
apply (case_tac "c=0", simp) 

1201 
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 

1202 
done 

1203 

1204 
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)" 

1205 
proof  

1206 
assume [simp]: "c\<noteq>0" 

1207 
have "(a = b/c) = (a*c = (b/c)*c)" 

1208 
by (simp add: field_mult_cancel_right) 

1209 
also have "... = (a*c = b)" 

1210 
by (simp add: divide_inverse mult_assoc) 

1211 
finally show ?thesis . 

1212 
qed 

1213 

1214 
lemma eq_divide_eq: 

1215 
"((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)" 

1216 
by (simp add: nonzero_eq_divide_eq) 

1217 

1218 
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)" 

1219 
proof  

1220 
assume [simp]: "c\<noteq>0" 

1221 
have "(b/c = a) = ((b/c)*c = a*c)" 

1222 
by (simp add: field_mult_cancel_right) 

1223 
also have "... = (b = a*c)" 

1224 
by (simp add: divide_inverse mult_assoc) 

1225 
finally show ?thesis . 

1226 
qed 

1227 

1228 
lemma divide_eq_eq: 

1229 
"(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)" 

1230 
by (force simp add: nonzero_divide_eq_eq) 

1231 

1232 
subsection{*Cancellation Laws for Division*} 

1233 

1234 
lemma divide_cancel_right [simp]: 

1235 
"(a/c = b/c) = (c = 0  a = (b::'a::{field,division_by_zero}))" 

1236 
apply (case_tac "c=0", simp) 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

1237 
apply (simp add: divide_inverse field_mult_cancel_right) 
14288  1238 
done 
1239 

1240 
lemma divide_cancel_left [simp]: 

1241 
"(c/a = c/b) = (c = 0  a = (b::'a::{field,division_by_zero}))" 

1242 
apply (case_tac "c=0", simp) 

14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

1243 
apply (simp add: divide_inverse field_mult_cancel_left) 
14288  1244 
done 
1245 

14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1246 
subsection {* Division and the Number One *} 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1247 

79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1248 
text{*Simplify expressions equated with 1*} 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1249 
lemma divide_eq_1_iff [simp]: 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1250 
"(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))" 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1251 
apply (case_tac "b=0", simp) 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1252 
apply (simp add: right_inverse_eq) 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1253 
done 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1254 

79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1255 

79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1256 
lemma one_eq_divide_iff [simp]: 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1257 
"(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))" 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1258 
by (simp add: eq_commute [of 1]) 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1259 

79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1260 
lemma zero_eq_1_divide_iff [simp]: 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1261 
"((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)" 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1262 
apply (case_tac "a=0", simp) 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1263 
apply (auto simp add: nonzero_eq_divide_eq) 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1264 
done 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1265 

79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1266 
lemma one_divide_eq_0_iff [simp]: 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1267 
"(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)" 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1268 
apply (case_tac "a=0", simp) 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1269 
apply (insert zero_neq_one [THEN not_sym]) 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1270 
apply (auto simp add: nonzero_divide_eq_eq) 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1271 
done 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1272 

79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1273 
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*} 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1274 
declare zero_less_divide_iff [of "1", simp] 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1275 
declare divide_less_0_iff [of "1", simp] 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1276 
declare zero_le_divide_iff [of "1", simp] 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1277 
declare divide_le_0_iff [of "1", simp] 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

1278 

14288  1279 

14293  1280 
subsection {* Ordering Rules for Division *} 
22542982bffd
moving some division theorems to Ring_and_Field
paulson 