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permissions  rwrr 
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(* Title: HOL/Ring_and_Field.thy 
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ID: $Id$ 
14770  3 
Author: Gertrud Bauer, Steven Obua, Lawrence C Paulson and Markus Wenzel 
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*) 
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14738  6 
header {* (Ordered) Rings and Fields *} 
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15229  8 
theory Ring_and_Field 
15140  9 
imports OrderedGroup 
15131  10 
begin 
14504  11 

14738  12 
text {* 
13 
The theory of partially ordered rings is taken from the books: 

14 
\begin{itemize} 

15 
\item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 

16 
\item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963 

17 
\end{itemize} 

18 
Most of the used notions can also be looked up in 

19 
\begin{itemize} 

14770  20 
\item \url{http://www.mathworld.com} by Eric Weisstein et. al. 
14738  21 
\item \emph{Algebra I} by van der Waerden, Springer. 
22 
\end{itemize} 

23 
*} 

14504  24 

14738  25 
axclass semiring \<subseteq> ab_semigroup_add, semigroup_mult 
26 
left_distrib: "(a + b) * c = a * c + b * c" 

27 
right_distrib: "a * (b + c) = a * b + a * c" 

14504  28 

14738  29 
axclass semiring_0 \<subseteq> semiring, comm_monoid_add 
14504  30 

14940  31 
axclass semiring_0_cancel \<subseteq> semiring_0, cancel_ab_semigroup_add 
32 

14738  33 
axclass comm_semiring \<subseteq> ab_semigroup_add, ab_semigroup_mult 
34 
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 

14940  51 
axclass comm_semiring_0_cancel \<subseteq> comm_semiring_0, cancel_ab_semigroup_add 
52 

53 
instance comm_semiring_0_cancel \<subseteq> semiring_0_cancel .. 

54 

14738  55 
axclass axclass_0_neq_1 \<subseteq> zero, one 
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zero_neq_one [simp]: "0 \<noteq> 1" 
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14738  58 
axclass semiring_1 \<subseteq> axclass_0_neq_1, semiring_0, monoid_mult 
14504  59 

14738  60 
axclass comm_semiring_1 \<subseteq> axclass_0_neq_1, comm_semiring_0, comm_monoid_mult (* previously almost_semiring *) 
61 

62 
instance comm_semiring_1 \<subseteq> semiring_1 .. 

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14738  64 
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  66 

14940  67 
axclass semiring_1_cancel \<subseteq> semiring_1, cancel_ab_semigroup_add 
68 

69 
instance semiring_1_cancel \<subseteq> semiring_0_cancel .. 

70 

14738  71 
axclass comm_semiring_1_cancel \<subseteq> comm_semiring_1, cancel_ab_semigroup_add (* previously semiring *) 
72 

14940  73 
instance comm_semiring_1_cancel \<subseteq> semiring_1_cancel .. 
74 

75 
instance comm_semiring_1_cancel \<subseteq> comm_semiring_0_cancel .. 

76 

14738  77 
axclass ring \<subseteq> semiring, ab_group_add 
78 

14940  79 
instance ring \<subseteq> semiring_0_cancel .. 
14504  80 

14738  81 
axclass comm_ring \<subseteq> comm_semiring_0, ab_group_add 
82 

83 
instance comm_ring \<subseteq> ring .. 

14504  84 

14940  85 
instance comm_ring \<subseteq> comm_semiring_0_cancel .. 
14738  86 

87 
axclass ring_1 \<subseteq> ring, semiring_1 

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14940  89 
instance ring_1 \<subseteq> semiring_1_cancel .. 
90 

14738  91 
axclass comm_ring_1 \<subseteq> comm_ring, comm_semiring_1 (* previously ring *) 
92 

93 
instance comm_ring_1 \<subseteq> ring_1 .. 

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14738  95 
instance comm_ring_1 \<subseteq> comm_semiring_1_cancel .. 
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14738  97 
axclass idom \<subseteq> comm_ring_1, axclass_no_zero_divisors 
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14738  99 
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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14940  103 
lemma mult_zero_left [simp]: "0 * a = (0::'a::semiring_0_cancel)" 
14738  104 
proof  
105 
have "0*a + 0*a = 0*a + 0" 

106 
by (simp add: left_distrib [symmetric]) 

107 
thus ?thesis 

108 
by (simp only: add_left_cancel) 

109 
qed 

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14940  111 
lemma mult_zero_right [simp]: "a * 0 = (0::'a::semiring_0_cancel)" 
14738  112 
proof  
113 
have "a*0 + a*0 = a*0 + 0" 

114 
by (simp add: right_distrib [symmetric]) 

115 
thus ?thesis 

116 
by (simp only: add_left_cancel) 

117 
qed 

118 

119 
lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0  b = 0)" 

120 
proof cases 

121 
assume "a=0" thus ?thesis by simp 

122 
next 

123 
assume anz [simp]: "a\<noteq>0" 

124 
{ assume "a * b = 0" 

125 
hence "inverse a * (a * b) = 0" by simp 

126 
hence "b = 0" by (simp (no_asm_use) add: mult_assoc [symmetric])} 

127 
thus ?thesis by force 

128 
qed 

129 

130 
instance field \<subseteq> idom 

131 
by (intro_classes, simp) 

132 

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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: 
14738  142 
"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  166 
by (simp add: left_distrib diff_minus 
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minus_mult_left [symmetric] minus_mult_right [symmetric]) 

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14738  169 
axclass pordered_semiring \<subseteq> semiring_0, pordered_ab_semigroup_add 
170 
mult_left_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b" 

171 
mult_right_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> a * c <= b * c" 

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14738  173 
axclass pordered_cancel_semiring \<subseteq> pordered_semiring, cancel_ab_semigroup_add 
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14940  175 
instance pordered_cancel_semiring \<subseteq> semiring_0_cancel .. 
176 

14738  177 
axclass ordered_semiring_strict \<subseteq> semiring_0, ordered_cancel_ab_semigroup_add 
178 
mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" 

179 
mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c" 

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14940  181 
instance ordered_semiring_strict \<subseteq> semiring_0_cancel .. 
182 

14738  183 
instance ordered_semiring_strict \<subseteq> pordered_cancel_semiring 
184 
apply intro_classes 

185 
apply (case_tac "a < b & 0 < c") 

186 
apply (auto simp add: mult_strict_left_mono order_less_le) 

187 
apply (auto simp add: mult_strict_left_mono order_le_less) 

188 
apply (simp add: mult_strict_right_mono) 

14270  189 
done 
190 

14738  191 
axclass pordered_comm_semiring \<subseteq> comm_semiring_0, pordered_ab_semigroup_add 
192 
mult_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b" 

14270  193 

14738  194 
axclass pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring, cancel_ab_semigroup_add 
14270  195 

14738  196 
instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring .. 
14270  197 

14738  198 
axclass ordered_comm_semiring_strict \<subseteq> comm_semiring_0, ordered_cancel_ab_semigroup_add 
199 
mult_strict_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" 

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14738  201 
instance pordered_comm_semiring \<subseteq> pordered_semiring 
202 
by (intro_classes, insert mult_mono, simp_all add: mult_commute, blast+) 

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14738  204 
instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring .. 
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14738  206 
instance ordered_comm_semiring_strict \<subseteq> ordered_semiring_strict 
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by (intro_classes, insert mult_strict_mono, simp_all add: mult_commute, blast+) 

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14738  209 
instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring 
210 
apply (intro_classes) 

211 
apply (case_tac "a < b & 0 < c") 

212 
apply (auto simp add: mult_strict_left_mono order_less_le) 

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apply (auto simp add: mult_strict_left_mono order_le_less) 

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done 
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14738  216 
axclass pordered_ring \<subseteq> ring, pordered_semiring 
14270  217 

14738  218 
instance pordered_ring \<subseteq> pordered_ab_group_add .. 
14270  219 

14738  220 
instance pordered_ring \<subseteq> pordered_cancel_semiring .. 
14270  221 

14738  222 
axclass lordered_ring \<subseteq> pordered_ring, lordered_ab_group_abs 
14270  223 

14940  224 
instance lordered_ring \<subseteq> lordered_ab_group_meet .. 
225 

226 
instance lordered_ring \<subseteq> lordered_ab_group_join .. 

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14738  228 
axclass axclass_abs_if \<subseteq> minus, ord, zero 
229 
abs_if: "abs a = (if (a < 0) then (a) else a)" 

14270  230 

14738  231 
axclass ordered_ring_strict \<subseteq> ring, ordered_semiring_strict, axclass_abs_if 
14270  232 

14738  233 
instance ordered_ring_strict \<subseteq> lordered_ab_group .. 
14270  234 

14738  235 
instance ordered_ring_strict \<subseteq> lordered_ring 
236 
by (intro_classes, simp add: abs_if join_eq_if) 

14270  237 

14738  238 
axclass pordered_comm_ring \<subseteq> comm_ring, pordered_comm_semiring 
14270  239 

14738  240 
axclass ordered_semidom \<subseteq> comm_semiring_1_cancel, ordered_comm_semiring_strict (* previously ordered_semiring *) 
241 
zero_less_one [simp]: "0 < 1" 

14270  242 

14738  243 
axclass ordered_idom \<subseteq> comm_ring_1, ordered_comm_semiring_strict, axclass_abs_if (* previously ordered_ring *) 
14270  244 

14738  245 
instance ordered_idom \<subseteq> ordered_ring_strict .. 
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14738  247 
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  251 
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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266 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

267 

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

268 
lemma less_add_iff2: 
14738  269 
"(a*e + c < b*e + d) = (c < (ba)*e + (d::'a::pordered_ring))" 
14272
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

270 
apply (simp add: diff_minus left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

271 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

272 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

273 

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

274 
lemma le_add_iff1: 
14738  275 
"(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

276 
apply (simp add: diff_minus left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

277 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

278 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

279 

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

280 
lemma le_add_iff2: 
14738  281 
"(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

282 
apply (simp add: diff_minus left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

283 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

284 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

285 

14270  286 
subsection {* Ordering Rules for Multiplication *} 
287 

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

288 
lemma mult_left_le_imp_le: 
14738  289 
"[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

290 
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

291 

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

292 
lemma mult_right_le_imp_le: 
14738  293 
"[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

294 
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

295 

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

296 
lemma mult_left_less_imp_less: 
14738  297 
"[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

298 
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

299 

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

300 
lemma mult_right_less_imp_less: 
14738  301 
"[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

302 
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

303 

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

304 
lemma mult_strict_left_mono_neg: 
14738  305 
"[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

306 
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

307 
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

308 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

309 

14738  310 
lemma mult_left_mono_neg: 
311 
"[b \<le> a; c \<le> 0] ==> c * a \<le> c * (b::'a::pordered_ring)" 

312 
apply (drule mult_left_mono [of _ _ "c"]) 

313 
apply (simp_all add: minus_mult_left [symmetric]) 

314 
done 

315 

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

316 
lemma mult_strict_right_mono_neg: 
14738  317 
"[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

318 
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

319 
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

320 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

321 

14738  322 
lemma mult_right_mono_neg: 
323 
"[b \<le> a; c \<le> 0] ==> a * c \<le> (b::'a::pordered_ring) * c" 

324 
apply (drule mult_right_mono [of _ _ "c"]) 

325 
apply (simp) 

326 
apply (simp_all add: minus_mult_right [symmetric]) 

327 
done 

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

328 

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

329 
subsection{* Products of Signs *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

330 

14738  331 
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

332 
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

333 

14738  334 
lemma mult_pos_le: "[ (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b ] ==> 0 \<le> a*b" 
335 
by (drule mult_left_mono [of 0 b], auto) 

336 

337 
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

338 
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

339 

14738  340 
lemma mult_pos_neg_le: "[ (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 ] ==> a*b \<le> 0" 
341 
by (drule mult_left_mono [of b 0], auto) 

342 

343 
lemma mult_pos_neg2: "[ (0::'a::ordered_semiring_strict) < a; b < 0 ] ==> b*a < 0" 

344 
by (drule mult_strict_right_mono[of b 0], auto) 

345 

346 
lemma mult_pos_neg2_le: "[ (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 ] ==> b*a \<le> 0" 

347 
by (drule mult_right_mono[of b 0], auto) 

348 

349 
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

350 
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

351 

14738  352 
lemma mult_neg_le: "[ a \<le> (0::'a::pordered_ring); b \<le> 0 ] ==> 0 \<le> a*b" 
353 
by (drule mult_right_mono_neg[of a 0 b ], auto) 

354 

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

355 
lemma zero_less_mult_pos: 
14738  356 
"[ 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

357 
apply (case_tac "b\<le>0") 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

358 
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

359 
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

360 
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

361 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

362 

14738  363 
lemma zero_less_mult_pos2: 
364 
"[ 0 < b*a; 0 < a] ==> 0 < (b::'a::ordered_semiring_strict)" 

365 
apply (case_tac "b\<le>0") 

366 
apply (auto simp add: order_le_less linorder_not_less) 

367 
apply (drule_tac mult_pos_neg2 [of a b]) 

368 
apply (auto dest: order_less_not_sym) 

369 
done 

370 

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

371 
lemma zero_less_mult_iff: 
14738  372 
"((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

373 
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

374 
apply (blast dest: zero_less_mult_pos) 
14738  375 
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

376 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

377 

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

378 
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

379 
assumption of an ordering. See @{text field_mult_eq_0_iff} below.*} 
14738  380 
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

381 
apply (case_tac "a < 0") 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

382 
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

383 
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

384 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

385 

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

386 
lemma zero_le_mult_iff: 
14738  387 
"((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

388 
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

389 
zero_less_mult_iff) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

390 

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

391 
lemma mult_less_0_iff: 
14738  392 
"(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

393 
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

394 
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

395 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

396 

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

397 
lemma mult_le_0_iff: 
14738  398 
"(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

399 
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

400 
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

401 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

402 

14738  403 
lemma split_mult_pos_le: "(0 \<le> a & 0 \<le> b)  (a \<le> 0 & b \<le> 0) \<Longrightarrow> 0 \<le> a * (b::_::pordered_ring)" 
404 
by (auto simp add: mult_pos_le mult_neg_le) 

405 

406 
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)" 

407 
by (auto simp add: mult_pos_neg_le mult_pos_neg2_le) 

408 

409 
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

410 
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

411 

14738  412 
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom} 
413 
theorems available to members of @{term ordered_idom} *} 

414 

415 
instance ordered_idom \<subseteq> ordered_semidom 

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

416 
proof 
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset

417 
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

418 
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

419 
qed 
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset

420 

14738  421 
instance ordered_ring_strict \<subseteq> axclass_no_zero_divisors 
422 
by (intro_classes, simp) 

423 

424 
instance ordered_idom \<subseteq> idom .. 

425 

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

426 
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

427 

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

428 
declare zero_neq_one [THEN not_sym, simp] 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

429 

14738  430 
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

431 
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

432 

14738  433 
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0" 
434 
by (simp add: linorder_not_le) 

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

435 

14738  436 
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0" 
437 
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

438 

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

439 
subsection{*More Monotonicity*} 
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{*Strict monotonicity in both arguments*} 
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<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

444 
apply (case_tac "c=0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

445 
apply (simp add: mult_pos) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

446 
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

447 
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

448 
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

449 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

450 

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

451 
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

452 
lemma mult_strict_mono': 
14738  453 
"[ 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

454 
apply (rule mult_strict_mono) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

455 
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

456 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

457 

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

458 
lemma mult_mono: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

459 
"[a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c] 
14738  460 
==> 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

461 
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

462 
apply (erule mult_left_mono, assumption) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

463 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

464 

14738  465 
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

466 
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

467 
apply (simp add: order_less_trans [OF zero_less_one]) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

468 
done 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

469 

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

470 
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

471 

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

472 
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

473 
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

474 

15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

475 
text{*These ``disjunction'' versions produce two cases when the comparison is 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

476 
an assumption, but effectively four when the comparison is a goal.*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

477 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

478 
lemma mult_less_cancel_right_disj: 
14738  479 
"(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

480 
apply (case_tac "c = 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

481 
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

482 
mult_strict_right_mono_neg) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

483 
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

484 
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

485 
linorder_not_le [symmetric, of a]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

486 
apply (erule_tac [!] notE) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

487 
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

488 
mult_right_mono_neg) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

489 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

490 

15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

491 
lemma mult_less_cancel_left_disj: 
14738  492 
"(c*a < c*b) = ((0 < c & a < b)  (c < 0 & b < (a::'a::ordered_ring_strict)))" 
493 
apply (case_tac "c = 0") 

494 
apply (auto simp add: linorder_neq_iff mult_strict_left_mono 

495 
mult_strict_left_mono_neg) 

496 
apply (auto simp add: linorder_not_less 

497 
linorder_not_le [symmetric, of "c*a"] 

498 
linorder_not_le [symmetric, of a]) 

499 
apply (erule_tac [!] notE) 

500 
apply (auto simp add: order_less_imp_le mult_left_mono 

501 
mult_left_mono_neg) 

502 
done 

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

503 

15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

504 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

505 
text{*The ``conjunction of implication'' lemmas produce two cases when the 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

506 
comparison is a goal, but give four when the comparison is an assumption.*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

507 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

508 
lemma mult_less_cancel_right: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

509 
fixes c :: "'a :: ordered_ring_strict" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

510 
shows "(a*c < b*c) = ((0 \<le> c > a < b) & (c \<le> 0 > b < a))" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

511 
by (insert mult_less_cancel_right_disj [of a c b], auto) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

512 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

513 
lemma mult_less_cancel_left: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

514 
fixes c :: "'a :: ordered_ring_strict" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

515 
shows "(c*a < c*b) = ((0 \<le> c > a < b) & (c \<le> 0 > b < a))" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

516 
by (insert mult_less_cancel_left_disj [of c a b], auto) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

517 

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

518 
lemma mult_le_cancel_right: 
14738  519 
"(a*c \<le> b*c) = ((0<c > a\<le>b) & (c<0 > b \<le> (a::'a::ordered_ring_strict)))" 
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

520 
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right_disj) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

521 

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

522 
lemma mult_le_cancel_left: 
14738  523 
"(c*a \<le> c*b) = ((0<c > a\<le>b) & (c<0 > b \<le> (a::'a::ordered_ring_strict)))" 
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

524 
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left_disj) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

525 

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

526 
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

527 
assumes less: "c*a < c*b" and nonneg: "0 \<le> c" 
14738  528 
shows "a < (b::'a::ordered_semiring_strict)" 
14377  529 
proof (rule ccontr) 
530 
assume "~ a < b" 

531 
hence "b \<le> a" by (simp add: linorder_not_less) 

532 
hence "c*b \<le> c*a" by (rule mult_left_mono) 

533 
with this and less show False 

534 
by (simp add: linorder_not_less [symmetric]) 

535 
qed 

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

536 

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

537 
lemma mult_less_imp_less_right: 
14738  538 
assumes less: "a*c < b*c" and nonneg: "0 <= c" 
539 
shows "a < (b::'a::ordered_semiring_strict)" 

540 
proof (rule ccontr) 

541 
assume "~ a < b" 

542 
hence "b \<le> a" by (simp add: linorder_not_less) 

543 
hence "b*c \<le> a*c" by (rule mult_right_mono) 

544 
with this and less show False 

545 
by (simp add: linorder_not_less [symmetric]) 

546 
qed 

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

547 

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

548 
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

549 
lemma mult_cancel_right [simp]: 
14738  550 
"(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

551 
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

552 
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

553 
simp add: linorder_neq_iff) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

554 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

555 

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

556 
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

557 
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

558 
lemma mult_cancel_left [simp]: 
14738  559 
"(c*a = c*b) = (c = (0::'a::ordered_ring_strict)  a=b)" 
560 
apply (cut_tac linorder_less_linear [of 0 c]) 

561 
apply (force dest: mult_strict_left_mono_neg mult_strict_left_mono 

562 
simp add: linorder_neq_iff) 

563 
done 

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

564 

15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

565 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

566 
subsubsection{*Special Cancellation Simprules for Multiplication*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

567 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

568 
text{*These also produce two cases when the comparison is a goal.*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

569 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

570 
lemma mult_le_cancel_right1: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

571 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

572 
shows "(c \<le> b*c) = ((0<c > 1\<le>b) & (c<0 > b \<le> 1))" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

573 
by (insert mult_le_cancel_right [of 1 c b], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

574 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

575 
lemma mult_le_cancel_right2: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

576 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

577 
shows "(a*c \<le> c) = ((0<c > a\<le>1) & (c<0 > 1 \<le> a))" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

578 
by (insert mult_le_cancel_right [of a c 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

579 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

580 
lemma mult_le_cancel_left1: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

581 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

582 
shows "(c \<le> c*b) = ((0<c > 1\<le>b) & (c<0 > b \<le> 1))" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

583 
by (insert mult_le_cancel_left [of c 1 b], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

584 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

585 
lemma mult_le_cancel_left2: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

586 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

587 
shows "(c*a \<le> c) = ((0<c > a\<le>1) & (c<0 > 1 \<le> a))" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

588 
by (insert mult_le_cancel_left [of c a 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

589 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

590 
lemma mult_less_cancel_right1: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

591 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

592 
shows "(c < b*c) = ((0 \<le> c > 1<b) & (c \<le> 0 > b < 1))" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

593 
by (insert mult_less_cancel_right [of 1 c b], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

594 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

595 
lemma mult_less_cancel_right2: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

596 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

597 
shows "(a*c < c) = ((0 \<le> c > a<1) & (c \<le> 0 > 1 < a))" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

598 
by (insert mult_less_cancel_right [of a c 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

599 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

600 
lemma mult_less_cancel_left1: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

601 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

602 
shows "(c < c*b) = ((0 \<le> c > 1<b) & (c \<le> 0 > b < 1))" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

603 
by (insert mult_less_cancel_left [of c 1 b], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

604 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

605 
lemma mult_less_cancel_left2: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

606 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

607 
shows "(c*a < c) = ((0 \<le> c > a<1) & (c \<le> 0 > 1 < a))" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

608 
by (insert mult_less_cancel_left [of c a 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

609 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

610 
lemma mult_cancel_right1 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

611 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

612 
shows "(c = b*c) = (c = 0  b=1)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

613 
by (insert mult_cancel_right [of 1 c b], force) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

614 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

615 
lemma mult_cancel_right2 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

616 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

617 
shows "(a*c = c) = (c = 0  a=1)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

618 
by (insert mult_cancel_right [of a c 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

619 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

620 
lemma mult_cancel_left1 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

621 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

622 
shows "(c = c*b) = (c = 0  b=1)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

623 
by (insert mult_cancel_left [of c 1 b], force) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

624 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

625 
lemma mult_cancel_left2 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

626 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

627 
shows "(c*a = c) = (c = 0  a=1)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

628 
by (insert mult_cancel_left [of c a 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

629 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

630 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

631 
text{*Simprules for comparisons where common factors can be cancelled.*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

632 
lemmas mult_compare_simps = 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

633 
mult_le_cancel_right mult_le_cancel_left 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

634 
mult_le_cancel_right1 mult_le_cancel_right2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

635 
mult_le_cancel_left1 mult_le_cancel_left2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

636 
mult_less_cancel_right mult_less_cancel_left 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

637 
mult_less_cancel_right1 mult_less_cancel_right2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

638 
mult_less_cancel_left1 mult_less_cancel_left2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

639 
mult_cancel_right mult_cancel_left 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

640 
mult_cancel_right1 mult_cancel_right2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

641 
mult_cancel_left1 mult_cancel_left2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

642 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

643 

14738  644 
text{*This list of rewrites decides ring equalities by ordered rewriting.*} 
15178  645 
lemmas ring_eq_simps = 
646 
(* mult_ac*) 

14738  647 
left_distrib right_distrib left_diff_distrib right_diff_distrib 
15178  648 
group_eq_simps 
649 
(* add_ac 

14738  650 
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 
15178  651 
diff_eq_eq eq_diff_eq *) 
14738  652 

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

653 
subsection {* Fields *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

654 

14288  655 
lemma right_inverse [simp]: 
656 
assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1" 

657 
proof  

658 
have "a * inverse a = inverse a * a" by (simp add: mult_ac) 

659 
also have "... = 1" using not0 by simp 

660 
finally show ?thesis . 

661 
qed 

662 

663 
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))" 

664 
proof 

665 
assume neq: "b \<noteq> 0" 

666 
{ 

667 
hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac) 

668 
also assume "a / b = 1" 

669 
finally show "a = b" by simp 

670 
next 

671 
assume "a = b" 

672 
with neq show "a / b = 1" by (simp add: divide_inverse) 

673 
} 

674 
qed 

675 

676 
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a" 

677 
by (simp add: divide_inverse) 

678 

15228  679 
lemma divide_self: "a \<noteq> 0 ==> a / (a::'a::field) = 1" 
14288  680 
by (simp add: divide_inverse) 
681 

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

682 
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

683 
by (simp add: divide_inverse) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

684 

15228  685 
lemma divide_self_if [simp]: 
686 
"a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)" 

687 
by (simp add: divide_self) 

688 

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

689 
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

690 
by (simp add: divide_inverse) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

691 

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

692 
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

693 
by (simp add: divide_inverse) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

694 

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

695 
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c" 
14293  696 
by (simp add: divide_inverse left_distrib) 
697 

698 

14270  699 
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement 
700 
of an ordering.*} 

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

701 
lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0  b = 0)" 
14377  702 
proof cases 
703 
assume "a=0" thus ?thesis by simp 

704 
next 

705 
assume anz [simp]: "a\<noteq>0" 

706 
{ assume "a * b = 0" 

707 
hence "inverse a * (a * b) = 0" by simp 

708 
hence "b = 0" by (simp (no_asm_use) add: mult_assoc [symmetric])} 

709 
thus ?thesis by force 

710 
qed 

14270  711 

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

712 
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

713 
lemma field_mult_cancel_right_lemma: 
14269  714 
assumes cnz: "c \<noteq> (0::'a::field)" 
715 
and eq: "a*c = b*c" 

716 
shows "a=b" 

14377  717 
proof  
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

718 
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

719 
by (simp add: eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

720 
thus "a=b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

721 
by (simp add: mult_assoc cnz) 
14377  722 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

723 

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

724 
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

725 
"(a*c = b*c) = (c = (0::'a::field)  a=b)" 
14377  726 
proof cases 
727 
assume "c=0" thus ?thesis by simp 

728 
next 

729 
assume "c\<noteq>0" 

730 
thus ?thesis by (force dest: field_mult_cancel_right_lemma) 

731 
qed 

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

732 

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

733 
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

734 
"(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

735 
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

736 

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

737 
lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)" 
14377  738 
proof 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

739 
assume ianz: "inverse a = 0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

740 
assume "a \<noteq> 0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

741 
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

742 
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

743 
finally have "1 = (0::'a::field)" . 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

744 
thus False by (simp add: eq_commute) 
14377  745 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

746 

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

747 

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

748 
subsection{*Basic Properties of @{term inverse}*} 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

749 

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

750 
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

751 
apply (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

752 
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

753 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

754 

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

755 
lemma inverse_nonzero_imp_nonzero: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

756 
"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

757 
apply (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

758 
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

759 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

760 

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

761 
lemma inverse_nonzero_iff_nonzero [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

762 
"(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

763 
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

764 

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

765 
lemma nonzero_inverse_minus_eq: 
14269  766 
assumes [simp]: "a\<noteq>0" shows "inverse(a) = inverse(a::'a::field)" 
14377  767 
proof  
768 
have "a * inverse ( a) = a *  inverse a" 

769 
by simp 

770 
thus ?thesis 

771 
by (simp only: field_mult_cancel_left, simp) 

772 
qed 

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

773 

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

774 
lemma inverse_minus_eq [simp]: 
14377  775 
"inverse(a) = inverse(a::'a::{field,division_by_zero})"; 
776 
proof cases 

777 
assume "a=0" thus ?thesis by (simp add: inverse_zero) 

778 
next 

779 
assume "a\<noteq>0" 

780 
thus ?thesis by (simp add: nonzero_inverse_minus_eq) 

781 
qed 

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

782 

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

783 
lemma nonzero_inverse_eq_imp_eq: 
14269  784 
assumes inveq: "inverse a = inverse b" 
785 
and anz: "a \<noteq> 0" 

786 
and bnz: "b \<noteq> 0" 

787 
shows "a = (b::'a::field)" 

14377  788 
proof  
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

789 
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

790 
by (simp add: inveq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

791 
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

792 
by simp 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

793 
thus "a = b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

794 
by (simp add: mult_assoc anz bnz) 
14377  795 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

796 

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

797 
lemma inverse_eq_imp_eq: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

798 
"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

799 
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

800 
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

801 
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

802 
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

803 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

804 

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

805 
lemma inverse_eq_iff_eq [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

806 
"(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

807 
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

808 

14270  809 
lemma nonzero_inverse_inverse_eq: 
810 
assumes [simp]: "a \<noteq> 0" shows "inverse(inverse (a::'a::field)) = a" 

811 
proof  

812 
have "(inverse (inverse a) * inverse a) * a = a" 

813 
by (simp add: nonzero_imp_inverse_nonzero) 

814 
thus ?thesis 

815 
by (simp add: mult_assoc) 

816 
qed 

817 

818 
lemma inverse_inverse_eq [simp]: 

819 
"inverse(inverse (a::'a::{field,division_by_zero})) = a" 

820 
proof cases 

821 
assume "a=0" thus ?thesis by simp 

822 
next 

823 
assume "a\<noteq>0" 

824 
thus ?thesis by (simp add: nonzero_inverse_inverse_eq) 

825 
qed 

826 

827 
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)" 

828 
proof  

829 
have "inverse 1 * 1 = (1::'a::field)" 

830 
by (rule left_inverse [OF zero_neq_one [symmetric]]) 

831 
thus ?thesis by simp 

832 
qed 

833 

15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

834 
lemma inverse_unique: 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

835 
assumes ab: "a*b = 1" 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

836 
shows "inverse a = (b::'a::field)" 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

837 
proof  
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

838 
have "a \<noteq> 0" using ab by auto 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

839 
moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

840 
ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

841 
qed 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

842 

14270  843 
lemma nonzero_inverse_mult_distrib: 
844 
assumes anz: "a \<noteq> 0" 

845 
and bnz: "b \<noteq> 0" 

846 
shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)" 

847 
proof  

848 
have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 

849 
by (simp add: field_mult_eq_0_iff anz bnz) 

850 
hence "inverse(a*b) * a = inverse(b)" 

851 
by (simp add: mult_assoc bnz) 

852 
hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 

853 
by simp 

854 
thus ?thesis 

855 
by (simp add: mult_assoc anz) 

856 
qed 

857 

858 
text{*This version builds in division by zero while also reorienting 

859 
the righthand side.*} 

860 
lemma inverse_mult_distrib [simp]: 

861 
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})" 

862 
proof cases 

863 
assume "a \<noteq> 0 & b \<noteq> 0" 

864 
thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_commute) 

865 
next 

866 
assume "~ (a \<noteq> 0 & b \<noteq> 0)" 

867 
thus ?thesis by force 

868 
qed 

869 

870 
text{*There is no slick version using division by zero.*} 

871 
lemma inverse_add: 

872 
"[a \<noteq> 0; b \<noteq> 0] 

873 
==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)" 

874 
apply (simp add: left_distrib mult_assoc) 

875 
apply (simp add: mult_commute [of "inverse a"]) 

876 
apply (simp add: mult_assoc [symmetric] add_commute) 

877 
done 

878 

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

879 
lemma inverse_divide [simp]: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

880 
"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

881 
by (simp add: divide_inverse mult_commute) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

882 

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

883 
lemma nonzero_mult_divide_cancel_left: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

884 
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

885 
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

886 
proof  
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

887 
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

888 
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

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

890 
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

891 
by (simp only: mult_ac) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

892 
also have "... = a * inverse b" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

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

894 
finally show ?thesis 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

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

896 
qed 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

897 

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

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

899 
"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

900 
apply (case_tac "b = 0") 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

901 
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

902 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

903 

14321  904 
lemma nonzero_mult_divide_cancel_right: 
905 
"[b\<noteq>0; c\<noteq>0] ==> (a*c) / (b*c) = a/(b::'a::field)" 

906 
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 

907 

908 
lemma mult_divide_cancel_right: 

909 
"c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})" 

910 
apply (case_tac "b = 0") 

911 
apply (simp_all add: nonzero_mult_divide_cancel_right) 

912 
done 

913 

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

914 
(*For ExtractCommonTerm*) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

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

916 
"(c*a) / (c*b) = 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

917 
(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

918 
by (simp add: mult_divide_cancel_left) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

919 

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

920 
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

921 
by (simp add: divide_inverse) 
14284
f1abe67c448a
reorganisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset

922 

15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

923 
lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)" 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

924 
by (simp add: divide_inverse mult_assoc) 
14288  925 

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

926 
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

927 
by (simp add: divide_inverse mult_ac) 
14288  928 

929 
lemma divide_divide_eq_right [simp]: 

930 
"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

931 
by (simp add: divide_inverse mult_ac) 
14288  932 

933 
lemma divide_divide_eq_left [simp]: 

934 
"(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

935 
by (simp add: divide_inverse mult_assoc) 
14288  936 

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

937 

15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

938 
subsubsection{*Special Cancellation Simprules for Division*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

939 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

940 
lemma mult_divide_cancel_left_if [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

941 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

942 
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

943 
by (simp add: mult_divide_cancel_left) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

944 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

945 
lemma mult_divide_cancel_right_if [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

946 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

947 
shows "(a*c) / (b*c) = (if c=0 then 0 else a/b)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

948 
by (simp add: mult_divide_cancel_right) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

949 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

950 
lemma mult_divide_cancel_left_if1 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

951 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

952 
shows "c / (c*b) = (if c=0 then 0 else 1/b)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

953 
apply (insert mult_divide_cancel_left_if [of c 1 b]) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

954 
apply (simp del: mult_divide_cancel_left_if) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

955 
done 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

956 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

957 
lemma mult_divide_cancel_left_if2 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

958 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

959 
shows "(c*a) / c = (if c=0 then 0 else a)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

960 
apply (insert mult_divide_cancel_left_if [of c a 1]) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

961 
apply (simp del: mult_divide_cancel_left_if) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

962 
done 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

963 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

964 
lemma mult_divide_cancel_right_if1 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

965 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

966 
shows "c / (b*c) = (if c=0 then 0 else 1/b)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

967 
apply (insert mult_divide_cancel_right_if [of 1 c b]) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

968 
apply (simp del: mult_divide_cancel_right_if) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

969 
done 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

970 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

971 
lemma mult_divide_cancel_right_if2 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

972 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

973 
shows "(a*c) / c = (if c=0 then 0 else a)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

974 
apply (insert mult_divide_cancel_right_if [of a c 1]) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

975 
apply (simp del: mult_divide_cancel_right_if) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

976 
done 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

977 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

978 
text{*Two lemmas for cancelling the denominator*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

979 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

980 
lemma times_divide_self_right [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

981 
fixes a :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

982 
shows "a * (b/a) = (if a=0 then 0 else b)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

983 
by (simp add: times_divide_eq_right) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

984 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

985 
lemma times_divide_self_left [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

986 
fixes a :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

987 
shows "(b/a) * a = (if a=0 then 0 else b)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

988 
by (simp add: times_divide_eq_left) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

989 

ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

990 

14293  991 
subsection {* Division and Unary Minus *} 
992 

993 
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==>  (a/b) = (a) / (b::'a::field)" 

994 
by (simp add: divide_inverse minus_mult_left) 

995 

996 
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==>  (a/b) = a / (b::'a::field)" 

997 
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right) 

998 

999 
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (a)/(b) = a / (b::'a::field)" 

1000 
by (simp add: divide_inverse nonzero_inverse_minus_eq) 

1001 

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

1002 
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

1003 
by (simp add: divide_inverse minus_mult_left [symmetric]) 
14293  1004 

1005 
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

1006 
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

1007 

14293  1008 

1009 
text{*The effect is to extract signs from divisions*} 

1010 
declare minus_divide_left [symmetric, simp] 

1011 
declare minus_divide_right [symmetric, simp] 

1012 

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

1013 
text{*Also, extract signs from products*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

1014 
declare minus_mult_left [symmetric, simp] 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

1015 
declare minus_mult_right [symmetric, simp] 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

1016 

14293  1017 
lemma minus_divide_divide [simp]: 
1018 
"(a)/(b) = a / (b::'a::{field,division_by_zero})" 

1019 
apply (case_tac "b=0", simp) 

1020 
apply (simp add: nonzero_minus_divide_divide) 

1021 
done 

1022 

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

1023 
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

1024 
by (simp add: diff_minus add_divide_distrib) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

1025 

14293  1026 

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

1027 
subsection {* Ordered Fields *} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1028 

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

1029 
lemma positive_imp_inverse_positive: 
14269  1030 
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

1031 
proof  
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1032 
have "0 < a * inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1033 
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

1034 
thus "0 < inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1035 
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

1036 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1037 

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

1038 
lemma negative_imp_inverse_negative: 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1039 
"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

1040 
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

1041 
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

1042 

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

1043 
lemma inverse_le_imp_le: 
14269  1044 
assumes invle: "inverse a \<le> inverse b" 
1045 
and apos: "0 < a" 

1046 
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

1047 
proof (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1048 
assume "~ b \<le> a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1049 
hence "a < b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1050 
by (simp add: linorder_not_le) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1051 
hence bpos: "0 < b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1052 
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

1053 
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

1054 
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

1055 
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

1056 
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

1057 
thus "b \<le> a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1058 
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

1059 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1060 

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

1061 
lemma inverse_positive_imp_positive: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1062 
assumes inv_gt_0: "0 < inverse a" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1063 
and [simp]: "a \<noteq> 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1064 
shows "0 < (a::'a::ordered_field)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

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

1066 
have "0 < inverse (inverse a)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1067 
by (rule positive_imp_inverse_positive) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1068 
thus "0 < a" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1069 
by (simp add: nonzero_inverse_inverse_eq) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1070 
qed 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1071 

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

1072 
lemma inverse_positive_iff_positive [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1073 
"(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

1074 
apply (case_tac "a = 0", simp) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1075 
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

1076 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1077 

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

1078 
lemma inverse_negative_imp_negative: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1079 
assumes inv_less_0: "inverse a < 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1080 
and [simp]: "a \<noteq> 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1081 
shows "a < (0::'a::ordered_field)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1082 
proof  
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1083 
have "inverse (inverse a) < 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1084 
by (rule negative_imp_inverse_negative) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1085 
thus "a < 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1086 
by (simp add: nonzero_inverse_inverse_eq) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1087 
qed 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1088 

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

1089 
lemma inverse_negative_iff_negative [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1090 
"(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

1091 
apply (case_tac "a = 0", simp) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1092 
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

1093 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1094 

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

1095 
lemma inverse_nonnegative_iff_nonnegative [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1096 
"(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

1097 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1098 

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

1099 
lemma inverse_nonpositive_iff_nonpositive [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1100 
"(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

1101 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1102 

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

1103 

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

1104 
subsection{*AntiMonotonicity of @{term inverse}*} 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1105 

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

1106 
lemma less_imp_inverse_less: 
14269  1107 
assumes less: "a < b" 
1108 
and apos: "0 < a" 

1109 
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

1110 
proof (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1111 
assume "~ inverse b < inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1112 
hence "inverse a \<le> inverse b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1113 
by (simp add: linorder_not_less) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1114 
hence "~ (a < b)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1115 
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

1116 
thus False 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1117 
by (rule notE [OF _ less]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1118 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1119 

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

1120 
lemma inverse_less_imp_less: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1121 
"[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

1122 
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

1123 
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

1124 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1125 

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

1126 
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

1127 
lemma inverse_less_iff_less [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1128 
"[0 < a; 0 < b] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1129 
==> (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

1130 
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

1131 

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

1132 
lemma le_imp_inverse_le: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1133 
"[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

1134 
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

1135 

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

1136 
lemma inverse_le_iff_le [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1137 
"[0 < a; 0 < b] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1138 
==> (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

1139 
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

1140 

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

1141 

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

1142 
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

1143 
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

1144 
lemma inverse_le_imp_le_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1145 
"[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

1146 
apply (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1147 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1148 
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

1149 
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

1150 
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

1151 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1152 

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

1153 
lemma less_imp_inverse_less_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1154 
"[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

1155 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1156 
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

1157 
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

1158 
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

1159 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1160 

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

1161 
lemma inverse_less_imp_less_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1162 
"[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

1163 
apply (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1164 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1165 
prefer 2 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1166 
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

1167 
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

1168 
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

1169 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1170 

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

1171 
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

1172 
"[a < 0; b < 0] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1173 
==> (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

1174 
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

1175 
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

1176 
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

1177 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1178 

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

1179 
lemma le_imp_inverse_le_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1180 
"[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

1181 
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

1182 

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

1183 
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

1184 
"[a < 0; b < 0] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1185 
==> (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

1186 
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

1187 

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

1188 

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

1189 
subsection{*Inverses and the Number One*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1190 

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

1191 
lemma one_less_inverse_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1192 
"(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

1193 
assume "0 < x" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1194 
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

1195 
show ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1196 
next 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1197 
assume notless: "~ (0 < x)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1198 
have "~ (1 < inverse x)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1199 
proof 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1200 
assume "1 < inverse x" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1201 
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

1202 
also have "... < 1" by (rule zero_less_one) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1203 
finally show False by auto 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1204 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1205 
with notless show ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1206 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1207 

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

1208 
lemma inverse_eq_1_iff [simp]: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1209 
"(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

1210 
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

1211 

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

1212 
lemma one_le_inverse_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1213 
"(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

1214 
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

1215 
eq_commute [of 1]) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1216 

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

1217 
lemma inverse_less_1_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1218 
"(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

1219 
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

1220 

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

1221 
lemma inverse_le_1_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1222 
"(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

1223 
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

1224 

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

1225 

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

1226 
subsection{*Division and Signs*} 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1227 

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

1228 
lemma zero_less_divide_iff: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1229 
"((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

1230 
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

1231 

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

1232 
lemma divide_less_0_iff: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1233 
"(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

1234 
(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

1235 
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

1236 

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

1237 
lemma zero_le_divide_iff: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1238 
"((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

1239 
(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

1240 
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

1241 

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

1242 
lemma divide_le_0_iff: 
14288  1243 
"(a/b \<le> (0::'a::{ordered_field,division_by_zero})) = 
1244 
(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

1245 
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

1246 

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

1247 
lemma divide_eq_0_iff [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1248 
"(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

1249 
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

1250 

14288  1251 

1252 
subsection{*Simplification of Inequalities Involving Literal Divisors*} 

1253 

1254 
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)" 

1255 
proof  

1256 
assume less: "0<c" 

1257 
hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)" 

1258 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

1259 
also have "... = (a*c \<le> b)" 

1260 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

1261 
finally show ?thesis . 

1262 
qed 

1263 

1264 

1265 
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)" 

1266 
proof  

1267 
assume less: "c<0" 

1268 
hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)" 

1269 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

1270 
also have "... = (b \<le> a*c)" 

1271 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

1272 
finally show ?thesis . 

1273 
qed 

1274 

1275 
lemma le_divide_eq: 

1276 
"(a \<le> b/c) = 

1277 
(if 0 < c then a*c \<le> b 

1278 
else if c < 0 then b \<le> a*c 

1279 
else a \<le> (0::'a::{ordered_field,division_by_zero}))" 

1280 
apply (case_tac "c=0", simp) 

1281 
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 