author  avigad 
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child 17085  5b57f995a179 
permissions  rwrr 
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(* Title: HOL/Ring_and_Field.thy 
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ID: $Id$ 
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Author: Gertrud Bauer, Steven Obua, Lawrence C Paulson, and Markus Wenzel, 
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with contributions by Jeremy Avigad 
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*) 
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14738  7 
header {* (Ordered) Rings and Fields *} 
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15229  9 
theory Ring_and_Field 
15140  10 
imports OrderedGroup 
15131  11 
begin 
14504  12 

14738  13 
text {* 
14 
The theory of partially ordered rings is taken from the books: 

15 
\begin{itemize} 

16 
\item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 

17 
\item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963 

18 
\end{itemize} 

19 
Most of the used notions can also be looked up in 

20 
\begin{itemize} 

14770  21 
\item \url{http://www.mathworld.com} by Eric Weisstein et. al. 
14738  22 
\item \emph{Algebra I} by van der Waerden, Springer. 
23 
\end{itemize} 

24 
*} 

14504  25 

14738  26 
axclass semiring \<subseteq> ab_semigroup_add, semigroup_mult 
27 
left_distrib: "(a + b) * c = a * c + b * c" 

28 
right_distrib: "a * (b + c) = a * b + a * c" 

14504  29 

14738  30 
axclass semiring_0 \<subseteq> semiring, comm_monoid_add 
14504  31 

14940  32 
axclass semiring_0_cancel \<subseteq> semiring_0, cancel_ab_semigroup_add 
33 

14738  34 
axclass comm_semiring \<subseteq> ab_semigroup_add, ab_semigroup_mult 
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 
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instance comm_semiring_0_cancel \<subseteq> semiring_0_cancel .. 

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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 *) 
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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: 
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"a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)" 
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by (simp add: left_distrib add_ac) 
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lemma minus_mult_left: " (a * b) = (a) * (b::'a::ring)" 
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apply (rule equals_zero_I) 
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apply (simp add: left_distrib [symmetric]) 
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done 
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lemma minus_mult_right: " (a * b) = a * (b::'a::ring)" 
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apply (rule equals_zero_I) 
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apply (simp add: right_distrib [symmetric]) 
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done 
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lemma minus_mult_minus [simp]: "( a) * ( b) = a * (b::'a::ring)" 
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by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemma minus_mult_commute: "( a) * b = a * ( b::'a::ring)" 
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by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemma right_diff_distrib: "a * (b  c) = a * b  a * (c::'a::ring)" 
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by (simp add: right_distrib diff_minus 
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minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemma left_diff_distrib: "(a  b) * c = a * c  b * (c::'a::ring)" 
14738  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 
207 
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 .. 

227 

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 
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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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15923  249 
lemmas linorder_neqE_ordered_idom = 
250 
linorder_neqE[where 'a = "?'b::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  254 
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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Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

267 
apply (simp add: diff_minus left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

268 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

269 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

270 

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

271 
lemma less_add_iff2: 
14738  272 
"(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

273 
apply (simp add: diff_minus left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

274 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

275 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

276 

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

277 
lemma le_add_iff1: 
14738  278 
"(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

279 
apply (simp add: diff_minus left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

280 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

281 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

282 

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

283 
lemma le_add_iff2: 
14738  284 
"(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

285 
apply (simp add: diff_minus left_distrib add_ac) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

286 
apply (simp add: compare_rls minus_mult_left [symmetric]) 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

287 
done 
5efbb548107d
Tidying of the integer development; towards removing the
paulson
parents:
14270
diff
changeset

288 

14270  289 
subsection {* Ordering Rules for Multiplication *} 
290 

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

291 
lemma mult_left_le_imp_le: 
14738  292 
"[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

293 
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

294 

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

295 
lemma mult_right_le_imp_le: 
14738  296 
"[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

297 
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

298 

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

299 
lemma mult_left_less_imp_less: 
14738  300 
"[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

301 
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

302 

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

303 
lemma mult_right_less_imp_less: 
14738  304 
"[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

305 
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

306 

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

307 
lemma mult_strict_left_mono_neg: 
14738  308 
"[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

309 
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

310 
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

311 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

312 

14738  313 
lemma mult_left_mono_neg: 
314 
"[b \<le> a; c \<le> 0] ==> c * a \<le> c * (b::'a::pordered_ring)" 

315 
apply (drule mult_left_mono [of _ _ "c"]) 

316 
apply (simp_all add: minus_mult_left [symmetric]) 

317 
done 

318 

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

319 
lemma mult_strict_right_mono_neg: 
14738  320 
"[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

321 
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

322 
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

323 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

324 

14738  325 
lemma mult_right_mono_neg: 
326 
"[b \<le> a; c \<le> 0] ==> a * c \<le> (b::'a::pordered_ring) * c" 

327 
apply (drule mult_right_mono [of _ _ "c"]) 

328 
apply (simp) 

329 
apply (simp_all add: minus_mult_right [symmetric]) 

330 
done 

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

331 

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

332 
subsection{* Products of Signs *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

333 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

334 
lemma mult_pos_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

335 
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

336 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

337 
lemma mult_nonneg_nonneg: "[ (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b ] ==> 0 \<le> a*b" 
14738  338 
by (drule mult_left_mono [of 0 b], auto) 
339 

340 
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

341 
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

342 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

343 
lemma mult_nonneg_nonpos: "[ (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 ] ==> a*b \<le> 0" 
14738  344 
by (drule mult_left_mono [of b 0], auto) 
345 

346 
lemma mult_pos_neg2: "[ (0::'a::ordered_semiring_strict) < a; b < 0 ] ==> b*a < 0" 

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

348 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

349 
lemma mult_nonneg_nonpos2: "[ (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 ] ==> b*a \<le> 0" 
14738  350 
by (drule mult_right_mono[of b 0], auto) 
351 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

352 
lemma mult_neg_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

353 
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

354 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

355 
lemma mult_nonpos_nonpos: "[ a \<le> (0::'a::pordered_ring); b \<le> 0 ] ==> 0 \<le> a*b" 
14738  356 
by (drule mult_right_mono_neg[of a 0 b ], auto) 
357 

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

358 
lemma zero_less_mult_pos: 
14738  359 
"[ 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

360 
apply (case_tac "b\<le>0") 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

361 
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

362 
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

363 
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

364 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

365 

14738  366 
lemma zero_less_mult_pos2: 
367 
"[ 0 < b*a; 0 < a] ==> 0 < (b::'a::ordered_semiring_strict)" 

368 
apply (case_tac "b\<le>0") 

369 
apply (auto simp add: order_le_less linorder_not_less) 

370 
apply (drule_tac mult_pos_neg2 [of a b]) 

371 
apply (auto dest: order_less_not_sym) 

372 
done 

373 

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

374 
lemma zero_less_mult_iff: 
14738  375 
"((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b  a < 0 & b < 0)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

376 
apply (auto simp add: order_le_less linorder_not_less mult_pos_pos 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

377 
mult_neg_neg) 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

378 
apply (blast dest: zero_less_mult_pos) 
14738  379 
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

380 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

381 

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

382 
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

383 
assumption of an ordering. See @{text field_mult_eq_0_iff} below.*} 
14738  384 
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

385 
apply (case_tac "a < 0") 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

386 
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

387 
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

388 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

389 

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

390 
lemma zero_le_mult_iff: 
14738  391 
"((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

392 
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

393 
zero_less_mult_iff) 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

394 

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

395 
lemma mult_less_0_iff: 
14738  396 
"(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

397 
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

398 
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

399 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

400 

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

401 
lemma mult_le_0_iff: 
14738  402 
"(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

403 
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

404 
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

405 
done 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

406 

14738  407 
lemma split_mult_pos_le: "(0 \<le> a & 0 \<le> b)  (a \<le> 0 & b \<le> 0) \<Longrightarrow> 0 \<le> a * (b::_::pordered_ring)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

408 
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos) 
14738  409 

410 
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)" 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

411 
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2) 
14738  412 

413 
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

414 
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

415 

14738  416 
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom} 
417 
theorems available to members of @{term ordered_idom} *} 

418 

419 
instance ordered_idom \<subseteq> ordered_semidom 

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

420 
proof 
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset

421 
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

422 
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

423 
qed 
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14398
diff
changeset

424 

14738  425 
instance ordered_ring_strict \<subseteq> axclass_no_zero_divisors 
426 
by (intro_classes, simp) 

427 

428 
instance ordered_idom \<subseteq> idom .. 

429 

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

430 
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

431 

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

432 
declare zero_neq_one [THEN not_sym, simp] 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

433 

14738  434 
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

435 
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

436 

14738  437 
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0" 
438 
by (simp add: linorder_not_le) 

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

439 

14738  440 
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0" 
441 
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

442 

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

443 
subsection{*More Monotonicity*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

444 

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

445 
text{*Strict monotonicity in both arguments*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

446 
lemma mult_strict_mono: 
14738  447 
"[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

448 
apply (case_tac "c=0") 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

449 
apply (simp add: mult_pos_pos) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

450 
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

451 
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

452 
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

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

454 

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

455 
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

456 
lemma mult_strict_mono': 
14738  457 
"[ 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

458 
apply (rule mult_strict_mono) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

459 
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

460 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

461 

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

462 
lemma mult_mono: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

463 
"[a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c] 
14738  464 
==> 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

465 
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

466 
apply (erule mult_left_mono, assumption) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

467 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

468 

14738  469 
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

470 
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

471 
apply (simp add: order_less_trans [OF zero_less_one]) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

472 
done 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

473 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

474 
lemma mult_less_le_imp_less: "(a::'a::ordered_semiring_strict) < b ==> 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

475 
c <= d ==> 0 <= a ==> 0 < c ==> a * c < b * d" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

476 
apply (subgoal_tac "a * c < b * c") 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

477 
apply (erule order_less_le_trans) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

478 
apply (erule mult_left_mono) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

479 
apply simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

480 
apply (erule mult_strict_right_mono) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

481 
apply assumption 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

482 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

483 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

484 
lemma mult_le_less_imp_less: "(a::'a::ordered_semiring_strict) <= b ==> 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

485 
c < d ==> 0 < a ==> 0 <= c ==> a * c < b * d" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

486 
apply (subgoal_tac "a * c <= b * c") 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

487 
apply (erule order_le_less_trans) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

488 
apply (erule mult_strict_left_mono) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

489 
apply simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

490 
apply (erule mult_right_mono) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

491 
apply simp 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

492 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

493 

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

494 
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

495 

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

496 
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

497 
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

498 

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

499 
text{*These ``disjunction'' versions produce two cases when the comparison is 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

500 
an assumption, but effectively four when the comparison is a goal.*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

501 

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

502 
lemma mult_less_cancel_right_disj: 
14738  503 
"(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

504 
apply (case_tac "c = 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

505 
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

506 
mult_strict_right_mono_neg) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

507 
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

508 
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

509 
linorder_not_le [symmetric, of a]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

510 
apply (erule_tac [!] notE) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

511 
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

512 
mult_right_mono_neg) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

513 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

514 

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

515 
lemma mult_less_cancel_left_disj: 
14738  516 
"(c*a < c*b) = ((0 < c & a < b)  (c < 0 & b < (a::'a::ordered_ring_strict)))" 
517 
apply (case_tac "c = 0") 

518 
apply (auto simp add: linorder_neq_iff mult_strict_left_mono 

519 
mult_strict_left_mono_neg) 

520 
apply (auto simp add: linorder_not_less 

521 
linorder_not_le [symmetric, of "c*a"] 

522 
linorder_not_le [symmetric, of a]) 

523 
apply (erule_tac [!] notE) 

524 
apply (auto simp add: order_less_imp_le mult_left_mono 

525 
mult_left_mono_neg) 

526 
done 

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

527 

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

528 

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

529 
text{*The ``conjunction of implication'' lemmas produce two cases when the 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

530 
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

531 

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

532 
lemma mult_less_cancel_right: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

533 
fixes c :: "'a :: ordered_ring_strict" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

534 
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

535 
by (insert mult_less_cancel_right_disj [of a c b], auto) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

536 

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

537 
lemma mult_less_cancel_left: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

538 
fixes c :: "'a :: ordered_ring_strict" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

539 
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

540 
by (insert mult_less_cancel_left_disj [of c a b], auto) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

541 

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

542 
lemma mult_le_cancel_right: 
14738  543 
"(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

544 
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

545 

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

546 
lemma mult_le_cancel_left: 
14738  547 
"(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

548 
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

549 

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

550 
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

551 
assumes less: "c*a < c*b" and nonneg: "0 \<le> c" 
14738  552 
shows "a < (b::'a::ordered_semiring_strict)" 
14377  553 
proof (rule ccontr) 
554 
assume "~ a < b" 

555 
hence "b \<le> a" by (simp add: linorder_not_less) 

556 
hence "c*b \<le> c*a" by (rule mult_left_mono) 

557 
with this and less show False 

558 
by (simp add: linorder_not_less [symmetric]) 

559 
qed 

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

560 

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

561 
lemma mult_less_imp_less_right: 
14738  562 
assumes less: "a*c < b*c" and nonneg: "0 <= c" 
563 
shows "a < (b::'a::ordered_semiring_strict)" 

564 
proof (rule ccontr) 

565 
assume "~ a < b" 

566 
hence "b \<le> a" by (simp add: linorder_not_less) 

567 
hence "b*c \<le> a*c" by (rule mult_right_mono) 

568 
with this and less show False 

569 
by (simp add: linorder_not_less [symmetric]) 

570 
qed 

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

571 

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

572 
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

573 
lemma mult_cancel_right [simp]: 
14738  574 
"(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

575 
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

576 
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

577 
simp add: linorder_neq_iff) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

578 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

579 

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

580 
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

581 
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

582 
lemma mult_cancel_left [simp]: 
14738  583 
"(c*a = c*b) = (c = (0::'a::ordered_ring_strict)  a=b)" 
584 
apply (cut_tac linorder_less_linear [of 0 c]) 

585 
apply (force dest: mult_strict_left_mono_neg mult_strict_left_mono 

586 
simp add: linorder_neq_iff) 

587 
done 

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

588 

15234
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 
subsubsection{*Special Cancellation Simprules for Multiplication*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

591 

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

592 
text{*These also produce two cases when the comparison is a goal.*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

593 

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

594 
lemma mult_le_cancel_right1: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

595 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

596 
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

597 
by (insert mult_le_cancel_right [of 1 c b], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

598 

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

599 
lemma mult_le_cancel_right2: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

600 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

601 
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

602 
by (insert mult_le_cancel_right [of a c 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

603 

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

604 
lemma mult_le_cancel_left1: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

605 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

606 
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

607 
by (insert mult_le_cancel_left [of c 1 b], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

608 

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

609 
lemma mult_le_cancel_left2: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

610 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

611 
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

612 
by (insert mult_le_cancel_left [of c a 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

613 

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

614 
lemma mult_less_cancel_right1: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

615 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

616 
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

617 
by (insert mult_less_cancel_right [of 1 c b], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

618 

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

619 
lemma mult_less_cancel_right2: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

620 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

621 
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

622 
by (insert mult_less_cancel_right [of a c 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

623 

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

624 
lemma mult_less_cancel_left1: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

625 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

626 
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

627 
by (insert mult_less_cancel_left [of c 1 b], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

628 

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

629 
lemma mult_less_cancel_left2: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

630 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

631 
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

632 
by (insert mult_less_cancel_left [of c a 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

633 

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

634 
lemma mult_cancel_right1 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

635 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

636 
shows "(c = b*c) = (c = 0  b=1)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

637 
by (insert mult_cancel_right [of 1 c b], force) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

638 

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

639 
lemma mult_cancel_right2 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

640 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

641 
shows "(a*c = c) = (c = 0  a=1)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

642 
by (insert mult_cancel_right [of a c 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

643 

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

644 
lemma mult_cancel_left1 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

645 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

646 
shows "(c = c*b) = (c = 0  b=1)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

647 
by (insert mult_cancel_left [of c 1 b], force) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

648 

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

649 
lemma mult_cancel_left2 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

650 
fixes c :: "'a :: ordered_idom" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

651 
shows "(c*a = c) = (c = 0  a=1)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

652 
by (insert mult_cancel_left [of c a 1], simp) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

653 

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

654 

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

655 
text{*Simprules for comparisons where common factors can be cancelled.*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

656 
lemmas mult_compare_simps = 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

657 
mult_le_cancel_right mult_le_cancel_left 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

658 
mult_le_cancel_right1 mult_le_cancel_right2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

659 
mult_le_cancel_left1 mult_le_cancel_left2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

660 
mult_less_cancel_right mult_less_cancel_left 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

661 
mult_less_cancel_right1 mult_less_cancel_right2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

662 
mult_less_cancel_left1 mult_less_cancel_left2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

663 
mult_cancel_right mult_cancel_left 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

664 
mult_cancel_right1 mult_cancel_right2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

665 
mult_cancel_left1 mult_cancel_left2 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

666 

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

667 

14738  668 
text{*This list of rewrites decides ring equalities by ordered rewriting.*} 
15178  669 
lemmas ring_eq_simps = 
670 
(* mult_ac*) 

14738  671 
left_distrib right_distrib left_diff_distrib right_diff_distrib 
15178  672 
group_eq_simps 
673 
(* add_ac 

14738  674 
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 
15178  675 
diff_eq_eq eq_diff_eq *) 
14738  676 

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

677 
subsection {* Fields *} 
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

678 

14288  679 
lemma right_inverse [simp]: 
680 
assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1" 

681 
proof  

682 
have "a * inverse a = inverse a * a" by (simp add: mult_ac) 

683 
also have "... = 1" using not0 by simp 

684 
finally show ?thesis . 

685 
qed 

686 

687 
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))" 

688 
proof 

689 
assume neq: "b \<noteq> 0" 

690 
{ 

691 
hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac) 

692 
also assume "a / b = 1" 

693 
finally show "a = b" by simp 

694 
next 

695 
assume "a = b" 

696 
with neq show "a / b = 1" by (simp add: divide_inverse) 

697 
} 

698 
qed 

699 

700 
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a" 

701 
by (simp add: divide_inverse) 

702 

15228  703 
lemma divide_self: "a \<noteq> 0 ==> a / (a::'a::field) = 1" 
14288  704 
by (simp add: divide_inverse) 
705 

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

706 
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

707 
by (simp add: divide_inverse) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

708 

15228  709 
lemma divide_self_if [simp]: 
710 
"a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)" 

711 
by (simp add: divide_self) 

712 

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

713 
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

714 
by (simp add: divide_inverse) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

715 

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

716 
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

717 
by (simp add: divide_inverse) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

718 

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

719 
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c" 
14293  720 
by (simp add: divide_inverse left_distrib) 
721 

722 

14270  723 
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement 
724 
of an ordering.*} 

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

725 
lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0  b = 0)" 
14377  726 
proof cases 
727 
assume "a=0" thus ?thesis by simp 

728 
next 

729 
assume anz [simp]: "a\<noteq>0" 

730 
{ assume "a * b = 0" 

731 
hence "inverse a * (a * b) = 0" by simp 

732 
hence "b = 0" by (simp (no_asm_use) add: mult_assoc [symmetric])} 

733 
thus ?thesis by force 

734 
qed 

14270  735 

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

736 
text{*Cancellation of equalities with a common factor*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

737 
lemma field_mult_cancel_right_lemma: 
14269  738 
assumes cnz: "c \<noteq> (0::'a::field)" 
739 
and eq: "a*c = b*c" 

740 
shows "a=b" 

14377  741 
proof  
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

742 
have "(a * c) * inverse c = (b * c) * inverse c" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

743 
by (simp add: eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

744 
thus "a=b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

745 
by (simp add: mult_assoc cnz) 
14377  746 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

747 

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

748 
lemma field_mult_cancel_right [simp]: 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

749 
"(a*c = b*c) = (c = (0::'a::field)  a=b)" 
14377  750 
proof cases 
751 
assume "c=0" thus ?thesis by simp 

752 
next 

753 
assume "c\<noteq>0" 

754 
thus ?thesis by (force dest: field_mult_cancel_right_lemma) 

755 
qed 

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

756 

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

757 
lemma field_mult_cancel_left [simp]: 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

758 
"(c*a = c*b) = (c = (0::'a::field)  a=b)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

759 
by (simp add: mult_commute [of c] field_mult_cancel_right) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

760 

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

761 
lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)" 
14377  762 
proof 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

763 
assume ianz: "inverse a = 0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

764 
assume "a \<noteq> 0" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

765 
hence "1 = a * inverse a" by simp 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

766 
also have "... = 0" by (simp add: ianz) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

767 
finally have "1 = (0::'a::field)" . 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

768 
thus False by (simp add: eq_commute) 
14377  769 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

770 

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

771 

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

772 
subsection{*Basic Properties of @{term inverse}*} 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

773 

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

774 
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::field)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

775 
apply (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

776 
apply (blast dest: nonzero_imp_inverse_nonzero) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

777 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

778 

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

779 
lemma inverse_nonzero_imp_nonzero: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

780 
"inverse a = 0 ==> a = (0::'a::field)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

781 
apply (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

782 
apply (blast dest: nonzero_imp_inverse_nonzero) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

783 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

784 

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

785 
lemma inverse_nonzero_iff_nonzero [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

786 
"(inverse a = 0) = (a = (0::'a::{field,division_by_zero}))" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

787 
by (force dest: inverse_nonzero_imp_nonzero) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

788 

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

789 
lemma nonzero_inverse_minus_eq: 
14269  790 
assumes [simp]: "a\<noteq>0" shows "inverse(a) = inverse(a::'a::field)" 
14377  791 
proof  
792 
have "a * inverse ( a) = a *  inverse a" 

793 
by simp 

794 
thus ?thesis 

795 
by (simp only: field_mult_cancel_left, simp) 

796 
qed 

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

797 

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

798 
lemma inverse_minus_eq [simp]: 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

799 
"inverse(a) = inverse(a::'a::{field,division_by_zero})" 
14377  800 
proof cases 
801 
assume "a=0" thus ?thesis by (simp add: inverse_zero) 

802 
next 

803 
assume "a\<noteq>0" 

804 
thus ?thesis by (simp add: nonzero_inverse_minus_eq) 

805 
qed 

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

806 

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

807 
lemma nonzero_inverse_eq_imp_eq: 
14269  808 
assumes inveq: "inverse a = inverse b" 
809 
and anz: "a \<noteq> 0" 

810 
and bnz: "b \<noteq> 0" 

811 
shows "a = (b::'a::field)" 

14377  812 
proof  
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

813 
have "a * inverse b = a * inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

814 
by (simp add: inveq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

815 
hence "(a * inverse b) * b = (a * inverse a) * b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

816 
by simp 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

817 
thus "a = b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

818 
by (simp add: mult_assoc anz bnz) 
14377  819 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

820 

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

821 
lemma inverse_eq_imp_eq: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

822 
"inverse a = inverse b ==> a = (b::'a::{field,division_by_zero})" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

823 
apply (case_tac "a=0  b=0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

824 
apply (force dest!: inverse_zero_imp_zero 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

825 
simp add: eq_commute [of "0::'a"]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

826 
apply (force dest!: nonzero_inverse_eq_imp_eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

827 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

828 

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

829 
lemma inverse_eq_iff_eq [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

830 
"(inverse a = inverse b) = (a = (b::'a::{field,division_by_zero}))" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

831 
by (force dest!: inverse_eq_imp_eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

832 

14270  833 
lemma nonzero_inverse_inverse_eq: 
834 
assumes [simp]: "a \<noteq> 0" shows "inverse(inverse (a::'a::field)) = a" 

835 
proof  

836 
have "(inverse (inverse a) * inverse a) * a = a" 

837 
by (simp add: nonzero_imp_inverse_nonzero) 

838 
thus ?thesis 

839 
by (simp add: mult_assoc) 

840 
qed 

841 

842 
lemma inverse_inverse_eq [simp]: 

843 
"inverse(inverse (a::'a::{field,division_by_zero})) = a" 

844 
proof cases 

845 
assume "a=0" thus ?thesis by simp 

846 
next 

847 
assume "a\<noteq>0" 

848 
thus ?thesis by (simp add: nonzero_inverse_inverse_eq) 

849 
qed 

850 

851 
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)" 

852 
proof  

853 
have "inverse 1 * 1 = (1::'a::field)" 

854 
by (rule left_inverse [OF zero_neq_one [symmetric]]) 

855 
thus ?thesis by simp 

856 
qed 

857 

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

858 
lemma inverse_unique: 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

859 
assumes ab: "a*b = 1" 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

860 
shows "inverse a = (b::'a::field)" 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

861 
proof  
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

862 
have "a \<noteq> 0" using ab by auto 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

863 
moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

864 
ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

865 
qed 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15010
diff
changeset

866 

14270  867 
lemma nonzero_inverse_mult_distrib: 
868 
assumes anz: "a \<noteq> 0" 

869 
and bnz: "b \<noteq> 0" 

870 
shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)" 

871 
proof  

872 
have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 

873 
by (simp add: field_mult_eq_0_iff anz bnz) 

874 
hence "inverse(a*b) * a = inverse(b)" 

875 
by (simp add: mult_assoc bnz) 

876 
hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 

877 
by simp 

878 
thus ?thesis 

879 
by (simp add: mult_assoc anz) 

880 
qed 

881 

882 
text{*This version builds in division by zero while also reorienting 

883 
the righthand side.*} 

884 
lemma inverse_mult_distrib [simp]: 

885 
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})" 

886 
proof cases 

887 
assume "a \<noteq> 0 & b \<noteq> 0" 

888 
thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_commute) 

889 
next 

890 
assume "~ (a \<noteq> 0 & b \<noteq> 0)" 

891 
thus ?thesis by force 

892 
qed 

893 

894 
text{*There is no slick version using division by zero.*} 

895 
lemma inverse_add: 

896 
"[a \<noteq> 0; b \<noteq> 0] 

897 
==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)" 

898 
apply (simp add: left_distrib mult_assoc) 

899 
apply (simp add: mult_commute [of "inverse a"]) 

900 
apply (simp add: mult_assoc [symmetric] add_commute) 

901 
done 

902 

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

903 
lemma inverse_divide [simp]: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

904 
"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

905 
by (simp add: divide_inverse mult_commute) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

906 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

907 
subsection {* Calculations with fractions *} 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

908 

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

909 
lemma nonzero_mult_divide_cancel_left: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

910 
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

911 
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

912 
proof  
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

913 
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

914 
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

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

916 
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

917 
by (simp only: mult_ac) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

918 
also have "... = a * inverse b" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

919 
by simp 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

920 
finally show ?thesis 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

921 
by (simp add: divide_inverse) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

922 
qed 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

923 

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

924 
lemma mult_divide_cancel_left: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

925 
"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

926 
apply (case_tac "b = 0") 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

927 
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

928 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

929 

14321  930 
lemma nonzero_mult_divide_cancel_right: 
931 
"[b\<noteq>0; c\<noteq>0] ==> (a*c) / (b*c) = a/(b::'a::field)" 

932 
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 

933 

934 
lemma mult_divide_cancel_right: 

935 
"c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})" 

936 
apply (case_tac "b = 0") 

937 
apply (simp_all add: nonzero_mult_divide_cancel_right) 

938 
done 

939 

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

940 
(*For ExtractCommonTerm*) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

941 
lemma mult_divide_cancel_eq_if: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

942 
"(c*a) / (c*b) = 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

943 
(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

944 
by (simp add: mult_divide_cancel_left) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

945 

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

946 
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

947 
by (simp add: divide_inverse) 
14284
f1abe67c448a
reorganisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents:
14277
diff
changeset

948 

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

949 
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

950 
by (simp add: divide_inverse mult_assoc) 
14288  951 

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

952 
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

953 
by (simp add: divide_inverse mult_ac) 
14288  954 

955 
lemma divide_divide_eq_right [simp]: 

956 
"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

957 
by (simp add: divide_inverse mult_ac) 
14288  958 

959 
lemma divide_divide_eq_left [simp]: 

960 
"(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

961 
by (simp add: divide_inverse mult_assoc) 
14288  962 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

963 
lemma add_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==> 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

964 
x / y + w / z = (x * z + w * y) / (y * z)" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

965 
apply (subgoal_tac "x / y = (x * z) / (y * z)") 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

966 
apply (erule ssubst) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

967 
apply (subgoal_tac "w / z = (w * y) / (y * z)") 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

968 
apply (erule ssubst) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

969 
apply (rule add_divide_distrib [THEN sym]) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

970 
apply (subst mult_commute) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

971 
apply (erule nonzero_mult_divide_cancel_left [THEN sym]) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

972 
apply assumption 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

973 
apply (erule nonzero_mult_divide_cancel_right [THEN sym]) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

974 
apply assumption 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

975 
done 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

976 

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

977 
subsubsection{*Special Cancellation Simprules for Division*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

978 

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

979 
lemma mult_divide_cancel_left_if [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

980 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

981 
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

982 
by (simp add: mult_divide_cancel_left) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

983 

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

984 
lemma mult_divide_cancel_right_if [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

985 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

986 
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

987 
by (simp add: mult_divide_cancel_right) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

988 

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

989 
lemma mult_divide_cancel_left_if1 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

990 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

991 
shows "c / (c*b) = (if c=0 then 0 else 1/b)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

992 
apply (insert mult_divide_cancel_left_if [of c 1 b]) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

993 
apply (simp del: mult_divide_cancel_left_if) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

994 
done 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

995 

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

996 
lemma mult_divide_cancel_left_if2 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

997 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

998 
shows "(c*a) / c = (if c=0 then 0 else a)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

999 
apply (insert mult_divide_cancel_left_if [of c a 1]) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1000 
apply (simp del: mult_divide_cancel_left_if) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1001 
done 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1002 

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

1003 
lemma mult_divide_cancel_right_if1 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1004 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1005 
shows "c / (b*c) = (if c=0 then 0 else 1/b)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1006 
apply (insert mult_divide_cancel_right_if [of 1 c b]) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1007 
apply (simp del: mult_divide_cancel_right_if) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1008 
done 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1009 

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

1010 
lemma mult_divide_cancel_right_if2 [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1011 
fixes c :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1012 
shows "(a*c) / c = (if c=0 then 0 else a)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1013 
apply (insert mult_divide_cancel_right_if [of a c 1]) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1014 
apply (simp del: mult_divide_cancel_right_if) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1015 
done 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1016 

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

1017 
text{*Two lemmas for cancelling the denominator*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1018 

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

1019 
lemma times_divide_self_right [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1020 
fixes a :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1021 
shows "a * (b/a) = (if a=0 then 0 else b)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1022 
by (simp add: times_divide_eq_right) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1023 

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

1024 
lemma times_divide_self_left [simp]: 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1025 
fixes a :: "'a :: {field,division_by_zero}" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1026 
shows "(b/a) * a = (if a=0 then 0 else b)" 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1027 
by (simp add: times_divide_eq_left) 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

1028 

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

1029 

14293  1030 
subsection {* Division and Unary Minus *} 
1031 

1032 
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==>  (a/b) = (a) / (b::'a::field)" 

1033 
by (simp add: divide_inverse minus_mult_left) 

1034 

1035 
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==>  (a/b) = a / (b::'a::field)" 

1036 
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right) 

1037 

1038 
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (a)/(b) = a / (b::'a::field)" 

1039 
by (simp add: divide_inverse nonzero_inverse_minus_eq) 

1040 

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

1041 
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

1042 
by (simp add: divide_inverse minus_mult_left [symmetric]) 
14293  1043 

1044 
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

1045 
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

1046 

14293  1047 

1048 
text{*The effect is to extract signs from divisions*} 

1049 
declare minus_divide_left [symmetric, simp] 

1050 
declare minus_divide_right [symmetric, simp] 

1051 

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

1052 
text{*Also, extract signs from products*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

1053 
declare minus_mult_left [symmetric, simp] 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

1054 
declare minus_mult_right [symmetric, simp] 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

1055 

14293  1056 
lemma minus_divide_divide [simp]: 
1057 
"(a)/(b) = a / (b::'a::{field,division_by_zero})" 

1058 
apply (case_tac "b=0", simp) 

1059 
apply (simp add: nonzero_minus_divide_divide) 

1060 
done 

1061 

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

1062 
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

1063 
by (simp add: diff_minus add_divide_distrib) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14377
diff
changeset

1064 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1065 
lemma diff_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==> 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1066 
x / y  w / z = (x * z  w * y) / (y * z)" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1067 
apply (subst diff_def)+ 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1068 
apply (subst minus_divide_left) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1069 
apply (subst add_frac_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1070 
apply simp_all 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1071 
done 
14293  1072 

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

1073 
subsection {* Ordered Fields *} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1074 

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

1075 
lemma positive_imp_inverse_positive: 
14269  1076 
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

1077 
proof  
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1078 
have "0 < a * inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1079 
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

1080 
thus "0 < inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1081 
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

1082 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1083 

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

1084 
lemma negative_imp_inverse_negative: 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1085 
"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

1086 
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

1087 
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

1088 

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

1089 
lemma inverse_le_imp_le: 
14269  1090 
assumes invle: "inverse a \<le> inverse b" 
1091 
and apos: "0 < a" 

1092 
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

1093 
proof (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1094 
assume "~ b \<le> a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1095 
hence "a < b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1096 
by (simp add: linorder_not_le) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1097 
hence bpos: "0 < b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1098 
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

1099 
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

1100 
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

1101 
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

1102 
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

1103 
thus "b \<le> a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1104 
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

1105 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1106 

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

1107 
lemma inverse_positive_imp_positive: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1108 
assumes inv_gt_0: "0 < inverse a" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1109 
and [simp]: "a \<noteq> 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1110 
shows "0 < (a::'a::ordered_field)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1111 
proof  
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1112 
have "0 < inverse (inverse a)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1113 
by (rule positive_imp_inverse_positive) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1114 
thus "0 < a" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1115 
by (simp add: nonzero_inverse_inverse_eq) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1116 
qed 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1117 

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

1118 
lemma inverse_positive_iff_positive [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1119 
"(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

1120 
apply (case_tac "a = 0", simp) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1121 
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

1122 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1123 

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

1124 
lemma inverse_negative_imp_negative: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1125 
assumes inv_less_0: "inverse a < 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1126 
and [simp]: "a \<noteq> 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1127 
shows "a < (0::'a::ordered_field)" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1128 
proof  
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1129 
have "inverse (inverse a) < 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1130 
by (rule negative_imp_inverse_negative) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1131 
thus "a < 0" 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1132 
by (simp add: nonzero_inverse_inverse_eq) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1133 
qed 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1134 

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

1135 
lemma inverse_negative_iff_negative [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1136 
"(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

1137 
apply (case_tac "a = 0", simp) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1138 
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

1139 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1140 

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

1141 
lemma inverse_nonnegative_iff_nonnegative [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1142 
"(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

1143 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1144 

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

1145 
lemma inverse_nonpositive_iff_nonpositive [simp]: 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1146 
"(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

1147 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1148 

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

1149 

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

1150 
subsection{*AntiMonotonicity of @{term inverse}*} 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

1151 

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

1152 
lemma less_imp_inverse_less: 
14269  1153 
assumes less: "a < b" 
1154 
and apos: "0 < a" 

1155 
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

1156 
proof (rule ccontr) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1157 
assume "~ inverse b < inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1158 
hence "inverse a \<le> inverse b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1159 
by (simp add: linorder_not_less) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1160 
hence "~ (a < b)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1161 
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

1162 
thus False 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1163 
by (rule notE [OF _ less]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1164 
qed 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1165 

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

1166 
lemma inverse_less_imp_less: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1167 
"[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

1168 
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

1169 
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

1170 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1171 

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

1172 
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

1173 
lemma inverse_less_iff_less [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1174 
"[0 < a; 0 < b] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1175 
==> (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

1176 
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

1177 

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

1178 
lemma le_imp_inverse_le: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1179 
"[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

1180 
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

1181 

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

1182 
lemma inverse_le_iff_le [simp]: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1183 
"[0 < a; 0 < b] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1184 
==> (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

1185 
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

1186 

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

1187 

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

1188 
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

1189 
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

1190 
lemma inverse_le_imp_le_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1191 
"[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

1192 
apply (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1193 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1194 
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

1195 
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

1196 
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

1197 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1198 

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

1199 
lemma less_imp_inverse_less_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1200 
"[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

1201 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1202 
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

1203 
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

1204 
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

1205 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1206 

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

1207 
lemma inverse_less_imp_less_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1208 
"[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

1209 
apply (rule classical) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1210 
apply (subgoal_tac "a < 0") 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1211 
prefer 2 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1212 
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

1213 
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

1214 
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

1215 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1216 

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

1217 
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

1218 
"[a < 0; b < 0] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1219 
==> (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

1220 
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

1221 
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

1222 
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

1223 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1224 

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

1225 
lemma le_imp_inverse_le_neg: 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1226 
"[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

1227 
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

1228 

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

1229 
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

1230 
"[a < 0; b < 0] 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

1231 
==> (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

1232 
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

1233 

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

1234 

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

1235 
subsection{*Inverses and the Number One*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1236 

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

1237 
lemma one_less_inverse_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1238 
"(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

1239 
assume "0 < x" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1240 
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

1241 
show ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1242 
next 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1243 
assume notless: "~ (0 < x)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1244 
have "~ (1 < inverse x)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1245 
proof 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1246 
assume "1 < inverse x" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1247 
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

1248 
also have "... < 1" by (rule zero_less_one) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1249 
finally show False by auto 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1250 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1251 
with notless show ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1252 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1253 

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

1254 
lemma inverse_eq_1_iff [simp]: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1255 
"(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

1256 
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

1257 

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

1258 
lemma one_le_inverse_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1259 
"(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

1260 
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

1261 
eq_commute [of 1]) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1262 

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

1263 
lemma inverse_less_1_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1264 
"(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

1265 
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

1266 

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

1267 
lemma inverse_le_1_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

1268 
"(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

1269 
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

1270 

14288  1271 
subsection{*Simplification of Inequalities Involving Literal Divisors*} 
1272 

1273 
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)" 

1274 
proof  

1275 
assume less: "0<c" 

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